I'd say that having a car is nice but not necessary, although that definitely depends on where you'll be working and living. My girlfriend went to Haaaaahvard school of public health for two years w/o a car and she was fine. Groceries were kind of a hassle, but the T (public transportation) goes pretty much everywhere.
The only thing I can think of, is that the T doesn't run 24 hours a day so if you're called out to the hospital late at night.... I guess you can always take a taxi.
I don't know about residency specifically, but I lived in Cambridge for 5 years (4 undergrad at MIT and the year after that) and I think it's very easy to live there without a car. What program are you entering and what hospital(s) will you be serving? You'll most likely find your places are quite accessable by the T (subway) or bus. There is almost no parking at MGH, but it has it's very own T stop.
If you are living in Boston proper, it's almost easier not to have a car. For what you'll pay in time and money to park it (pronouced pAArk), plus limited free time, renting one for a day or two or long weekend once a month is likely to be cheaper. Also, in order to register a car in Massachusettes, you HAVE to have proof of insurance (more $$).
All that said, I had a car for 3 years....but (1) I personally wasn't paying for it or to insure it, (2) parking was a hassle, (3) there were more than a few parking tickets, (4) it was towed more than once for street cleaning -- the second time it took 3 or 4 days before we realized it was gone, and (4) while I was working in the city, I drove it mostly to the laundry mat and the grocery store. However, prior to that I did all my grocery shopping by T and it's one of my fondest memories.