Are you supposed to store a baked cake in the refrigerator?

not necessarily. It depends on the type of cake. If it's a cake that consists of main ingredients that you would typically store in the fridge (ie cream or cheese) then yes, definitely keep it in the fridge. If it's a regular baked cake, like a chocolate cake, then you could leave it out. Put it on a plate or platter, whatever you may have, and cover it with foil or saran wrap. Keep it in a relatively dry place and it should be fine for at least a week if you keep it covered.