I suppose we have all had this experience, unless you don't shower or have hair.
My shower is flush with the floor but a plunger was not enough and it was not possible to stick a coat hanger down the hole.
Finally I removed the floor plate and found there was a sort of upside down cup to keep the foul oder from coming up (I think). This cup was easily removed along with the "sewer rat" (hair ball) attached to it. The cup just needed to be filled with water again and clicked back into place.
Of course not everyone's shower is like this, but I wanted to mention another alternative in case you also have a walk-in shower.
Problem was solved but needs to be repeated every few months.
As Coolio recommended, use gloves if you have them.
CH-iao