That's just plain stupid. Take a look at the map of the city I live in. It's 30 miles in diameter, and there's nothing but other cities outside those city limits. There simply is no means by which a mass transit system could replace the road system in my city, as there is absolutely no "center" that people go to--- everyone lives somewhere else and goes to a different place to work. You probably live in one of those "cities" with 300K people that can easily be served by two light rail lines and a dozen buses. When you have a greater metropolitan area that's home to 12 million plus people that spans a dozen city entities in two counties, mass transit becomes a much bigger problem than can be solved by an idiotic handwave of "just ban cars from city limits".
That's perfectly doable. It would cost a lot, but I bet it would cost less then everyone buying those cars.
I won't even begin to address the issue of what you consider "cars" and what constitutes a legitimately necessary vehicle. No... I will. Do you expect supermarkets to get food deliveries by bus? Is the plumber going to bring tools and 10-foot lengths of copper pipe to your house on the subway? Are old people who can barely walk expected to somehow drag 30 pounds of groceries home a kilometer from the nearest transit station? No, I'm guessing you'd suggest some sort of "permit" system that'd allow certain "special" classes of people to have personal vehicles... and like any such system, those with money would be able to game it and drive as they please. So what you're really suggesting is that poor people should be banned from driving in the city.
Now thats the actual legitimate argument for cars. They are in fact very useful for transporting heavy stuff, and often that sort of thing is a many-many relation. I suspect that with a good enough system you could do this just fine, but it would definitely be tricky.