The options aren't just single family houses and high-rises. You relax the zoning requirements that raise the lot size required per resident. Lot size minimums, boundary minimums between houses, number of floors, parking minimums, and minimum unit size. Re-legalize single-room occupancy units, so that people who can only afford the bare minimum (room with a lock) can buy that and not be homeless.
Or if those technical requirements that make cheap housing illegal aren't palatable, then just add in much more mixed frame construction. It's already fairly popular, I've seen a bunch of buildings under construction like this. It's basically 1-2 floors of steel or concrete frame for commercial use, along with 3-5 floors of residential wood framed units on top of that. Replace low-density commercial with that mid-density mixed-use, add in mid-rise townhouses, and you've got plenty of room to have five times as many folks in Berkeley.
Of course, the real problem is that housing is too regional for this to make a big difference. It's a huge cost to Berkeley, and what it does is relieve pressure in the greater metropolitan area as people shift their commutes to take advantage of this.