Yes, some of the so-called "super sentos" do offer 24-hour accommodation and they can serve as impromptu cheap hotels. The super-sentos as well as net cafes are quickly being seen as alternatives to capsule hotels especially for women.
The super-sentos will usually charge a surcharge for staying past 11pm or midnight and so the total fee will be between Y2000-5000 depending on how fancy the place is and its location (central Tokyo is of course much more expensive).
You would stay the night in their relaxation rooms which feature reclineable chairs (much like first-class seats on an airline). All of your belongings are in the locker room, so you don't have to worry about theft. There are magazines and pop-out tvs to entertain you (if you know Japanese). Some relaxation rooms are mixed sex, some are single-sex, and some places have both. You wear the yukata or sento-provided pajamas while inside.
The larger super-sentos have in-house restaurants so you really never have to leave.
The only problem with the super-sentos for visitors is that they don't have luggage storage. You really can only come in with what will fit in your locker -- which is just about the same size as a gym locker in the USA -- so not big enough for a roller suitcase. But if you stored your luggage at the coin locker in the station, it's certainly an option.
Also, most don't allow multi-day stays. They kick you out from 9a - 11a (for example), so that they can clean the facilities. You can take a long walk (with your belongings) and check back in, but it's not the most convenient for that.
Still, I keep it as an option for when I just really need a place to crash for one night.