Housed in creaky 1920s apartment building, the District Hotel gives off something of a Europe-on-$25-a-day vibe—small, dark, run-down in places, a fair share of students, and not much in the way of amenities—but it's a real bargain for the neighborhood. The big perk is the Downtown location: within walking distance are two Metro stations, the White House, restaurants, a grocery store, and lots...