I have one of those wash bags for the washing.. i have a small plastic storage tub that i put some hay into and leave inside ready to use..A lot of shop bought hays that come in sealed plastic bags are dry enough to be kept in their original packaging. I get a lot of hay from hay for pets - this hay isn't damp but seems to have a natural "freshness" to it and needs to be stored in a breathable container.
You can store hay in anything that will let the air get to it . I have heard of the following being used ...
an old duvet
large cardboard box
wicker laundry basket.
I use these ...
not the cheapest option, but I have found a 4.5 kilo bale of INGS fits in nicely