Most students rent off campus in the adjoining neighborhoods (Inner Sunset, Haight/Ashbury, Cole Valley). UCSF does have student housing, much of which is for families and is up on this enormous hill (10 minute shuttle ride from Parnassus campus). You can still get a spot there as a single student if you get on the lists early enough.
There is some more UCSF housing opening very shortly at Mission Bay, the new research campus, but of course that is of little use unless you're in the graduate phase of your training at a lab there as it's very far away from the Parnassus campus.
Rent off-campus runs pretty high and most places are relatively old (but charming?). You can get a studio all to yourself for maybe 1000/mo, share a 2 bdrm/1 bath apt. for 700-800/mo. Of course, there's a range, and you can find places for cheaper if you get lucky and there are some ritzier places if you want to pay more.
Check out housing.ucsf.edu for UCSF options and craigslist.com for rentals (search Inner Sunset neighborhood).