Air-tight is tricky. It’s actually better to keep it in a dry place in a container that’s rather airy. Otherwise the tiniest moisture in the hay combined with bacteria that naturally exists in earth may cause really bad toxins to form in the hay if there’s no oxygen – f ex botulinum bacteria. As you know botulinum is perhaps the most toxic toxin in the world. It has no taste and no smell and heat destroys it, but since you dobn’t cook hay, there is a risk. There are horses that die of botulism.
In a non-air-tight but still closed plastic bag, the hay may rot or get moldy if it’s just a little bit too moist when it’s bagged. I had that happen recently- I ought to have opened the bag right when I bought it, but I bought 3 at a time and left one bag in my mum’s house. When I did open it it didn’t smell right, I had to throw it all out. Rot and mold may also kill a bunny.
So I really recommend airy storage of hay. Cardboard boxes or paper-bags or opened plastic bags, and to store it indoors where the humidity is low.