Most students live in Foxhall or in Glover Park, both are about 10-15 minute walks to campus.
I would recommend living near campus. It's just convenient. Most people I know who live far from campus don't go to class and only show up when they need to. It's definitely nice having a bunch of other med students around for social reasons. It's true that Gtown itself does get a little boring. However, you don't really have a ton of free time, and when you do, it's easy to get to Dupont or Adams Morgan (by public transportation, GUTS, or even cabs are reasonable if you have a few people). It's more expensive near campus, but I think it is the better option personally. If you just have a bit of time to chill with friends, living far will just be annoying since pretty much everyone lives close together. If you want to avoid other students at all costs, by all means move far away. But most people hang with other med students, so living close is a very good thing. I would definitely live close first year, and then maybe consider moving away second year if you really want to.
Another note about GUTS. They run about every two hours or so on Saturdays and not at all on Sundays. Not always the convenient option.
You should be able to find housing in Foxhall or Glover Park. I didn't find my place until June, so there's time. Keep looking, but also wait for the social network to open up. People post listings on there specifically because they want med students. Or, you can find roomies who already have a house, or find a group to help make the search easier.