The thing is, you would have to deal with it as if you were making good hay -- that means getting it REALLY DRY before storing (and then storing in a place and manner that will keep it from getting humid again, which is a recipe for mold). This is not nearly as easy as you might think, especially for fine stuff like lawn clippings.
I would say leave them on the lawn (do not bag) for a couple days -- honestly, if I were going to do it, I would get a good accurate kitchen scale, and actually weigh them to determine how moisture content changes so that I *knew* I was waiting til they were dry enough -- then rake up and store in loose piles in a completely enclosed, dirt- or wood-floored building. Preferably up on windowscreen- or burlap-covered pallets. (A concrete- or gravel-floored building would probably be ok as long as you will use the 'lawn hay' up by midwinter -- but after that, you will start running into humidity and mold problems.)
If you don't want to have to rake 'em off the lawn, then you will need a large area to spread the bagged clippings out so that they can dry. Perhaps a paved driveway. But it will take a While for them to dry, and you need to be as careful as described above not to try to store them til they are really ready. With real hay, it needs to have a moisture content of no more than 15-20% max -- I would err on the side of caution with lawn clippings, since their fine texture and tendency to pack down in storage will make them extra prone to mold.
Honestly it sounds like a terrific amount of work for not much benefit, to me. If you really want to harvest your own hay for bedding, I'd suggest using a string trimmer on a field of tall grass that has grown up all summer, and dry and store *that* -- you will at least get some reasonable volume of material out of it
Good luck,
Pat
