My bathroom sink gets a lot of hair in it (even though I never brush my hair over the sink, I have to at least style it there, as the only mirror in the apartment is right over that sink). Chemicals don't work, vinegar and baking soda don't work, hot water doesn't work. The hair gets wrapped around just about everything and won't budge. I finally went out and bought a plastic tool called a Zip-it that has little sharp prongs sticking out from a flexible strip of plastic. You stick it down your drain and pull it back up again, and it catches the hair so you can pull it out.
It did end up popping the drain stopper off the little metal thing that makes it go up and down, but it didn't break it. Since I live in an apartment complex, we're going to have the maintenance people come and fix that, but I don't think it's all that complicated to fix yourself.