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I live in the 2nd hottest neighborhood in the city. You can rent an apartment within 4 blocks of me, tomorrow, for $800 a month, and you'd have many choices. If you were willing to live further out or in a less prestigious neighborhood you could trade up on size, too.
[+] Tokyo is very much not monolithically tall buildings. I live in a 45 story highrise; less than 25 meters away there exists single-family housing and 3 story apartments. The average density is pretty high, though -- here in Meguro it is about 2.5x SF's citywide (50k people per square mile vs 20k).
SF would need to invest tens or hundreds of billions of dollars in transit in order to make Tokyo style densities work. Just digging out the sides of Embarcadero station to allow two-sided boarding for BART is projected to cost a billion dollars.