Look at the posting in a previous thread titled Large Roll of Batting two postings below this one. Several members posted photos as to how they store batting rolls. Don't know if any of those would keep the cat hair or even the cats away from the batting roll but it's worth a look. Cats are wonderful pets but when they are still youngish, they seem to be able to get anywhere they want to, high, low or inbetween.

I keep my roll of batting in the corner of my craft room. It has a door on it, but often my cats go in there. They aren't really interested in the batting. When I need to cut batting, I vacuum the family room floor and roll it out. The batting is 120 inches wide, folded in half in the roll... so it's quite large when it's full.

How about a cover? Something simple...fabric the length of the roll and just a little wider. Attach Velcro every few inches as needed. When you're not using just cover it up. If you use dense enough fabric it won't matter if kitty scratches on it now and then. When you need it, just zip open the Velcro and remove the cover. Just a thought. Might work.

I use a table cloth (Vinyl) to cover mine, Buy off season ones on sale . Fold in half sew across
bottom and up the side turn wrong side out so it doesn't stick to the flannel backing. Slides in and out easy.
If you want to use pole on frame leave both ends open .

I keep the heavy plastic that it is wrapped in. Actually mine comes in two plastic bags, so I keep the interior one. I also have a cat and everything is open except my closet which is where the batting lives propped up against the corner. Not elegant, but it works.

My rolls of batting are stored in 2 places. The one I use most often is on the bar below the LA. The others are wool, bamboo and 120" wide and are hung from the ceiling. I put large binder clips on each end to keep from unrolling. With the clips, I can cut a length by myself. DH is also willing to help. Because the room is used mainly for LA, it stays faily dust free, so they aren't covered.

I have mine on the top of PVC shelving unit my Husband made. The unit is 20 in. wide so I can put 2 rolls of 10 yds side by side and 1 on top of them. I would like to hang a roll from ceiling and slightly in front of the back roller. But, I never have my sewing room neat enough for my hubby to work in it. He's a neat person, I'm not. that's why we bought a 3 bedroom house. Tinker

My MIL mounted extra closet rod hangers above the bi-fold closet door in her sewing room and just runs the rod through her next roll of batting. She keeps a bungee cord strapped to it- other wise the weight will cause it to unroll. She just undoes the bungee cord, unrolls the amount she needs, cuts it off and puts the bungee cord back on. Very easy and it’s up out of the way.

No longarm here, but I do keep my rolls of batting stored up on end in a guest room closet, as others have said they do. It's the same room where I use the bed to lay out quilts and sandwich them, so I just take the roll out and cut of what i need, then put the rest back in the closet. They stand on their plastic that they came in.

I buy my batting by the bolt so had to find a better place to put it for use. Since my sewing room is in the basement and my rafters above are open to me I decided to add brackets and a pole to hang the batting above my cutting table. I also keep a pair of clamps to hold my very long metal ruler to the cutting table for when I go to cut the batting. My cutting table is 4 x 8 so I can't cut it across from one side..........reason for the clamps to keep the ruler from moving on me. Here's pics of my setup.