The biggest problem that I have had is that the small pieces of hay will vanish into the lawn. To prevent this I would let it dry for several hours and then rake it up before it was entirely dry.
I was careful to keep the drying "hay" pile light and fluffy: once it was entirely dry I could put it into a clean garbage can and THEN moderately pack it down.
Then, when I wanted it for chicken bedding, it would still be good smelling and healthy.
The chickens loved the dry grass clippings to scratch at, but it would break down in perhaps 3 months time, and so in the Fall I would need enough stored to replace it in, say, January.
It takes a bit of time to do it, and I would hate to try to store enough for a small flock of sheep, but I would not hesitate to store enough for rabbits.