I buy hay by the bale and store it in my garage- the bale is wrapped in a large cotton blanket- and the bale is in our wheelbarrow so that it is off the floor. The hay stays clean and dry this way.
I know they make actually bags that are meant to hit a bale of hay (horse owners use them)- but I think when I looked the bags were about $20 and I only pay $4 a bale- and I already had the old cotton blanket. The wheelbarrow or something else would still be necessary to keep off the ground to keep it clean, dry and inaccessible to other pests.
Do you have someone local that you can buy a bale of hay from? I happened to be driving past a farm and noticed a “Hay for Sale” sign with a phone number. I wrote down the number and have gotten 2 bales so far from him. Ironically enough I think I am the only one that buys ONE bale at a time. He is a super nice guy- and I am glad that he is willing to sell only one bale at a time to me.
Just be sure to ask what type of hay it is and what cutting it is. That way you know what to ask for or look for in the future. Another place to find someone that will sell a single bale of hay might be at a local Farmers Market- they most likely would not have the hay there- but would be able to make arrangements.