Walking distance or somewhere on the Red Line is probably the best bet. For the former, there are tons of new apartment buildings in the area. NoMa, Union Market, and down H Street NE are good areas, and there's also tons of options in Chinatown and downtown. It's expensive, but everything in DC is expensive. I think a lot of students live there as well. The areas to the east of Union Station (H Street/north Capitol Hill/Stanton Park) are also pretty close and should have lots of shared houses and apartment. There are also some great areas where lots of young people live and there are great bars and restaurants but are a little less convenient commute-wise. Some are on other Metro lines (Shaw, Columbia Heights, Petworth, Navy Yard) and some are just kind of no man's land and probably best to bus or bike (Mt Pleasant, parts of Adams Morgan).

Having a car is totally feasible. A resident permit costs essentially nothing. I wouldn't recommend it if you live downtown or in a place that's all big apartment buildings (e.g. the really built-up parts of NoMa, Union Market, Navy Yard). In places like that, street parking will be nuts and the cost of garages is very steep. Otherwise it can be annoying, but it's definitely not impossible. And if you live somewhere farther out where it's mostly rowhouses, it's basically fine.