We know we can create lifts that do underground and then connect entry points and exit points by tunnel networks. We already have very miniature things like this.
What the video shows is a vast network of underground tunnels with a tram system for personal vehicles and some public transit connecting different points in the city or connecting other cities. I can kinda see why you might do this and it sounds like a fun ride.
But even a layman can see this is going to be incredibly expensive and you have to wonder why would build this ahead of other types of mass transit. I'd like to see a business plan. Until then, I'm a sceptic.
The problems of zoning alone make this a massive undertaking. Much of the counter-criticism in this thread is misdirected. Very few people are doubting Musk's technical ability to pull this off, they are suggesting it worn't work for practical reasons. You don't have to be a rocket scientist to understand those, just have to have lived in a city, or simply lived.