Actually, it's more feasible than you might think.
I have a friend in Blacksburg who's lived his whole life without a car (he's about 10 years older than me). He's held a regular job in the entire time (walks to work normally, rides the bus in bad weather). In town he walks where ever he needs to go or takes a bus, and he lives reasonably close to a grocery store. He likes to travel and probably travels more than most people (usually flies or goes by train and knows how to get good deals on his trips).
It is easier to explain this to someone from here :)
I did it a little bit myself even in Houston, but that is mostly for Housotn traffic. When we lived in the north part of town in the suburbs, I used to take the HOV lane city bus to office for 5 years. Then when we moved inside the loop, it was no longer viable, because now I have to drop kids to school before I go to work.
Yes, it is possible even in Houston, but typically if you are young and single.