Even on non-prime real-estate (which does that exist in SF?), I don't think you could hit your $20 price point with the amount of housekeeping needed for a bed and shower. Even if you could, after a week your staff might hate you. :-)
Capsule hotels don't usually have attached showers though do they? It's more of a communal thing, which maybe that would work just fine for what mmanfrin was thinking of. Though historically Americans aren't as accustomed to communal bathing (well except for gyms) as the Japanese.
To be fair, they say Japanese love hotels allow single guests (https://en.wikivoyage.org/wiki/Japan#Love_hotels) and capsule hotels have showers (https://en.wikivoyage.org/wiki/Japan#Capsule_hotels).
I don't doubt, though, that even in Japan there's a niche for specifically targeted establishments of the kind described by grandparent.