Acceleration/Deceleration Tunnel Length
Let's assume for a moment that Musk is able to bring down the cost of tunneling by a factor of 10 (a heady assumption which Alon doesn't believe). According to his own FAQ, this would bring the cost of tunneling to about $100 million per mile. Which would be relatively impressive, although Seattle manages to build tunnels for $300 million per mile, including systems (track, electrification, stations, etc). So how much is he proposing to tunnel? Further down his thread, he says he'd have thousands of small-stations designed to bring you close to your destination, and that each would be the size of a parking spot. That's the size of the station on the surface. What's more important, and more expensive, is the size of the station underground.
His plan is to have the car skate slot cars (which, I'll point out, don't exactly exist) travel at a speed of 125 mph. There will be trunk lines which will ferry a steady stream of cars at this speed. But to gain any efficiency in these trunks, car-skates will have to accelerate to speed before they can merge in to traffic, much like a freeway has long ramps to let cars accelerate to speed to merge in. Of course, highway merges and acceleration zones are generally designed for 60 or 70 mph, and design guidelines are up to 2000 feet.
How long would the car-skate acceleration zones be? Well, let's assume an acceleration speed of 10 miles per hour per second. This is equivalent to zero to 60 in six seconds: not quite the acceleration of a Tesla, but quite fast (about 0.5 G): a speed at which you definitely wouldn't want standees (subway trains max out at about 3 mph/s for this reason). But assuming everyone is seated, the top potential acceleration may be faster, but a margin for error may be needed to move in to a gap in traffic (like when you enter a freeway, you don't just gun your engine to top speed, you match the speed of other traffic and then move in to an open spot). If you start from a dead stop and accelerate to 125 miles per hour, you'll cover about 1/4 miles before you're at top speed. (This is a linear relationship, so if you did limit it to 3 mph/s, you'd be tripling the length of the acceleration tunnels.)
So take that as the length of tunnel required to depart each station. And double it. Because you have to decelerate at a similar rate from the other direction. So now you have to build 1000 stations each with half a mile of tunnel just to arrive and depart before you've built a single mile of main line tunnel. That 500 miles of tunnel or, by Musk's rosy calculations, $50 billion dollars. It increases the size of each station by a factor of 100—at least—and increases the size where it's most expensive to build: underground. Theoretically, a single deceleration zone could serve multiple stations: car-skates could slow down off a trunk line and then proceed at a slower speed to the final car-skate elevator (every time I write this sentence I chuckle at the outlandishness, and also Mitt Romney). But then you're building just as many access tunnels (although maybe you keep your mainline tunnel costs down but just making the trip times longer). Or you could decelerate on the main line, but then you may slow cars behind you. (As we'll see later, though, the number of cars in the system is not the rate limiting factor, so this might be attainable and save a bit of time. However, the movie does not show this.)
Ability to Serve Multiple Locations
Take another look at the video. The car sits in a plaza, people get in to the car, the car descends down some sort of elevator and then it reaches the slot car stage and accelerates away. The elevator must use hydraulics, because there's nowhere for the pulleys for a counterweight to hide above ground. Hydraulic elevators don't require overhead equipment, but they're also not very fast, running at a maximum of 200 feet per minute, although existing car elevators are really slow (and given the size and complexity of the elevator shown here, even 100 fpm seems quite fast). To clear utilities and other existing infrastructure and keep construction costs down, these tunnels will likely be at least 50 feet underground, so that's a minimum of 30 seconds of elevator time, but probably closer to a minute. Then the car-skate has to leave the elevator, another car-skate has to come on to it, and then after going up, the people have to get on and off. So they cycle time of a single trip at each station will be about two minutes, maybe 1:30 at best, if the elevator is fast and no one holds the door for their late-running friend.
If we assume two minutes, that means that a maximum of 30 car-skates per hour can serve each station. So while Musk promises that you'll be able to get to any number of 1000 stations, in reality, you'll only be able to get to about 30 within any hour, and only one every two minutes. Let's assume you want service to your destination within 10 minutes: on one of the next five car-skates to depart. There would be a 1-in-200 chance that a car-skate would be going your way. Even if there were five car-skates departing simultaneously from each station, your chances would be pretty slim. And that would drive up the costs of building the system in the first place; no longer would it be "just one parking spot."
Capacity
You'd be lucky to get half of this theoretical maximum, so it's probably more like 100 people per space. With 1000 stations, that means you'd have a maximum capacity of about 100,000 people per hour. That's actually pretty high, although on par with a medium-sized subway system; Chicago, for instance, has 1460 subway cars each capable of holding about 80 passengers, or about 120,000 people at any given time. (Musk's trip times would be shorter, so the rate limiting factor would not be how many people he could move through the system, but how he could get them in and out.) This, of course, assumes a balanced load, i.e. that every space would have similar numbers of people going up and down at all times of day, which isn't exactly the case. So you'd have a lot of full downward trips in residential areas in the morning, but once people got off downtown, the car-skate elevators would be empty going down for their next trip.
But remember our costs! Just building the station access tunnels would cost $50 billion. So to serve a CTA-level of ridership (half a million daily passengers) you'd need to spend $100,000 per rider in capital costs. Plus, these costs don't scale: to double the number of riders you'd have to double the number of access tunnels (or, perhaps more expensively, build more car-skate elevators). You haven't even built a single mile of main travel tunnel (or anything else) and you already have the largest single construction project in history.
Merging Tunnel Boring Machines
Let's say you solve the $50 billion dollar question above (good luck). Now you're ready to build your system. Put those TBMs in the ground and go! Not so fast. Tunnel boring machines don't exactly merge well. Go back and watch Musk's video. Once the car-skate accelerates to speed, it then merges in to the existing tunnel. But that's not how TBMs work. When you run a TBM, you build a pit to put it in the ground, dig a tunnel, break through, and pull it out of a separate extraction pit at the end. (In the case of the Chunnel, two of the TBMs were just aimed down and left in place, with the tunnel completed above them.) But connecting two tunnels isn't easy or cheap, you have to basically do it by hand: the connection and egress tunnels for the Chunnel were an engineering challenge.
Now, imagine two tunnels merging. TBMs are designed to leave a round, sealed tunnel behind them. But to merge tunnels, you'd have to merge to almost-parallel tubes. So for a long distance, several hundred feet if not longer, you'd have to cut open the side of one TBM and the allow the other to cut merging in to it, or to hand-dig or hand-mine the entire length (this might not be too bad with hard material, but in clay, sand, mud or anything beyond hard rock it's, uh, not easy). That drives up costs exponentially. And you'd have to do this for every single station. So remember our $50 billion number? It might have an extra zero on the end by the time all is said and done.
The video shows car-skates zooming through an open cavern with supports in between the tunnels. Maybe this is shown for effect. But in reality, this kind of infrastructure can't be built. It certainly can't be built with tunnel boring machines, and would probably be impossible without razing the entire city and starting anew. And while tunnels may be relatively safe during earthquakes (he's starting this in LA, after all), jacking the entire city up on to supports probably wouldn't pass muster.
Other Tunnel Costs
So far, we've addressed the cost to build the acceleration and deceleration tunnels to get people in and out of Musk's system and arrived at a cost of $100 billion (I'm doubling the $50 billion cost, and I think that's more than generous) to serve a moderately-sized city. Of course, there are a few other costs involved. First, you need the trunk line tunnels, and because you have to be able to get from any one tunnel to any other tunnel, you'll need a lot of them, a lot of merges, and a lot of ramps between them. You'll need either long, gradually curved tunnels to connect them (because at 125 mph, you can't go around sharp curves, you'd need about a half mile curve radius to keep G forces at comfortable levels) or lots of deceleration and acceleration zones to slow down and make sharper curves (which require many more merges which aren't cheap). In a subway, changing from one direction to another is easy, either the whole train slows down and goes through a junction and around a curve, or people get off of one train, go down a flight of stairs, and get on to another. When you have to create a system with dozens of tunnels all linked together, it gets a bit trickier.
And that's just what's underground. You also need 1000 stations above ground. Back to the video, note that the car-skate itself is about the size of a parking space, but there's also a plaza with people milling about. You can't build either of these easily in an existing right-of-way, because existing rights-of-way are home to utilities underground which make digging down difficult and costly. So to save money, you'd have to buy several hundred square feet of property all over the city. You also need to build emergency egress locations (subways have these quite frequently), ventilation shafts, fire suppression systems and the like. None of these is huge, but you need a lot of them: 1000 stations, and probably as many vent shafts and stairwells and the like. Each would have property acquisition costs, construction costs, permitting issues, utility relocations, and certainly unforeseen complications.
More stations = more cost
Our example has 1000 stations, but is this enough? A city like Boston or Chicago has about 10,000 people per square mile; New York has more, LA has fewer. Even so, this would mean interconnecting 100 square miles with stations and tubes. If you have 1000 stations in 100 square miles, you'd have about a 31-by-31 grid, or a station about every third of a mile. This would provide good accessibility: putting everyone within about a 7 or 8 minute walk of a station. Of course, it wouldn't be appreciably better than, say, Chicago's gridded bus network, and the cost would be somewhat higher. Plus, in dense areas you'd need more stations because of the limited capacity for each of about 100 passengers per hour, but you'd still have to build the same density further out to keep it in a short walking distance from most homes.
But that's only one million people. Consider Los Angeles: 12 million people in 1600 square miles. To serve Los Angeles, you'd need to build a system 16 times larger than we've calculated. 16,000 station access tubes alone would cost $1.6 trillion. The rest of the infrastructure would probably double that (add in engineering costs, repair depots, etc., and it probably goes higher). So to build this level of infrastructure in LA would be $3 trillion or more, or $250,000 per person. That's one sixth of the GDP of the US, four times LA's annual GDP, and at a per-person rate, five times the per capita income.
But, yeah, the video sure looks cool.
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