One of the possible ways to simulate gravity in outer space is to have a rotating spaceship, so that the centrifugal force experienced provides a gravity-like force.

My question is: shouldn't this only work when our feet are touching the floor of the spaceship? Only in that case the floor is providing a contact force to balance the centrifugal force.

If we jumped, there is no gravity within the spacecraft so what is it that would make us come back down?

Also: imagine we had a shower: what would make the water fall down?

If you jumped "straight up", you would still have a horizontal component of velocity (relative to a nonrotating frame), so you would still end up coming "back down". 

Likewise, the shower water is moving horizontally in a nonrotating frame, which makes it collide with the floor eventually (since the floor is curving upwards in the nonrotating frame). But to a person on the ship, it looks as if the water was moving downwards, rather than the floor (and you) moving upwards.

More dangerous would be if you were to try to run in the opposite direction of the rotation; if you ran fast enough, you would eventually find that you had become weightless. This would also mean that your feet would no longer be touching the ground, the world would be spinning underneath you, and you'd have no way of getting back down again. 

Fortunately, since the air is also moving due to the rotation, the "wind" would eventually "slow you down" (technically it would actually speed you up) and you would eventually regain "gravity" and fall to the ground.
