I live about 5 blocks from Georgetown and have a few friends who are MS-II, almost MS-IIIs there. First, it depends how close to campus you are living...in some areas the street parking is really tight and you have to have a DC parking permit (done by wards) to park on the street for more than two hours. You could walk/take the bus to get groceries. There is also a University bus from the campus to the closest metro (Dupont Circle) so that is convenient. You may need to get a car your 3rd and 4th year depending on where you are rotating. Washington Hospital Center is one of the main hospitals where GU students rotate and it is accesible by the metro and then a shuttle bus.
So, I think you could get by without a car, but it could be an inconvience at times- then again having a car could also be an inconvience at times. I personally would feel lost without my car, but I also lead a different life then I do when I start med school.
If you want to PM with where you are living I can tell you how the parking situation is...and whatever else you might want to know.