Guest houses (also called gaijin houses) can be rented by the day, week, or month. They are like boarding houses, and some offer private rooms (a little more expensive, of course, than the usual facilities).
Then, there are places like LeoPalace21 that offer fully furnished apartments with no guarantor needed. With them, you pay EVERYTHING up front, whether you stay a week or a year, plus a service charge for cleaning. Nice places from what I hear (and see on the ads), but who has the money to pay for a year up front?