I did my undergrad at U of M in Ann Arbor in 97-2001, and I did NOT have a car. This wasn't ever an issue until I needed to go grocery shopping or to the airport... And whenever I wanted to go to a rock concert in Detroit, I usually had to buy two tickets so I could give the other one to whomever gave me a ride.
Commuter Express ran a shuttle to and from the airport, but, back then, it was $45 each way (from what I can remember). You can always get a taxi, and it's not impossible to get to the grocery stores on the bus. I have to say, the bus system in Ann Arbor is pretty good for a smallish city. You can actually get by on public transportation alone. You can even take Amtrak or Greyhound to and from Chicago and Detroit pretty easily, and the rides are not unreasonably long (I've done both, many times).
Parking is very, very difficult to find and the police officers are out to make money by ticketing you wherever and whenever they can. Ann Arbor's notorious for this! While you might wish you had a car sometimes, (while you're hauling ten bags of groceries on the bus, for example) you will definitely not miss the hassle (and the extra $135 your landlord is charging!). You can always make friends with people who have cars and carpool to the grocery store. I think there is even a Trader Joe's pretty close to campus and definitely close to a bus line.
Ask me if you have any more questions about Ann Arbor!
(I should point out, though, that some people who did have cars were probably darn grateful that they had them!)