The majority of SF is two story suburbs (and as the song goes, they are made out of ticky-tacky). Height wise, it is underbuilt, and that's why SF has such a low population density despite every piece of land being covered in these rickety little bungalows. The bungalow owners fight tooth and nail against denser multistory construction because they realize that the housing crunch is the only possible way that their shacks can have such high prices.

You don't need to build on any green areas to double the housing capacity of SF.

Compared to actual dense cities in the world, SF is nearly empty.

SF has 1/4 of the population density of Manhattan, the city it is often compared to, and about 1/6 of the densest city in the world (Manila in the Philippines). I wonder where all those extra people are being put if SF has capped out?