If you want to live in San Francisco proper without paying a terrifying amount, try looking at some neighborhoods that are not the same places where everyone wants to live (Mission, SOMA, Lower Haight, North Beach, NOPA, etc). The Inner Richmond or Sunset are good picks if you don't mind having a 20-30 minute commute. Still expensive, but not as bad as the aforementioned neighborhoods. Those can easily be $2k per person if you are signing a new lease and not moving in with someone with rent control.

If you want some substantial space for $2k or under, I would look at the East Bay exclusively.

Be prepared to be shocked either way.