I keep bales in wheely bins outside. I pick up the bales using hay bale bags and just put the whole thig in a spare wheely bin in the shade. The hay isn't in direct contact with the plastic, so it can still breathe.
As long as it's dry and out of direct sun so it doesn't sweat, it will be OK - so off the floor in a shed or garage, etc. It really only spoils if it gets damp.