If you can get to it, you can remove the trap and check out the pipes.
Drains have an S thing on them that keeps water in it (called a trap). This keeps sewer gases from coming up and making your place stink (or explode). As one might expect, this trap is also a great place for hair and everything else to gather together and stop water from going down. If you can get to this thing from below your drain (probably the floor below your bathroom), then check it out. It's very easy if you have these modern plastic pipes, although a tad disgusting.
Otherwise, try the hanger-hook idea or the snake before Draino. Draino has chemicals that can cause burns, so it's better to try physical removal methods before chemical.
BTW: If you are renting, this isn't your problem, but your landlord's.
