(My comment is redundant with some of the others but I wrote it before reading them. I figure it's helpful to see concurring opinions.)
I think that you will find such short term rentals difficult but not impossible to come by. As a fallback, though it isn't a very exciting option, there are various hotel-but-with-kitchens businesses that specialize in short-term rentals.
The rental market is laid fairly bare on Craigslist for the SF Bay Area. There you might find, closer to the time of your travel, temporary sublet or roommate opportunities.
You don't make clear what you mean by "magic". I'll take two guesses that you mean either crazy Bay Area culture or insane Bay Area business networking opportunities. For culture, consider Berkeley (in the flats, not the hills) or, within The City (S.F.) consider the Castro, Haight, N. Beach, or SOMA neighborhoods (given your Kevlar aversion, I'd suggest avoiding the Tenderloin). For business, I am guessing a bit but consider SOMA or, for a complete twist into a kind of bland suburban hell (but close to a lot of VCs and start-ups) consider heading south on the peninsula to Mtn. View or Palo Alto or somewhere around there.
I'd caution against most of Oakland. In the hills you'll be isolated from action. In many parts of the flats, you'll wish for your Kevlar. There are excellent and peaceful and vibrant places to live in Oakland but if you are picking from afar, don't count on your ability to tell them from the not so good places. Similarly, sadly, avoid Richmond.
If you time your visit for early summer, after the Universities end their spring semester, the chances of finding nice short-term sublets or roommate situations goes way up.
I think some of the price quotes you got below are not impossible but you should expect higher.
Be very careful not to get scammed when negotiating a deal from afar (of course).
-t