For me, I have shoe boxes organized by colors for pieces that are between something like 3 inches and a FQ. Smaller than that, or really wacky shaped, go into a box for crumb blocks.

For yardage pieces of fabric, I ruler fold on a 6.5 inch ruler. Then fold the piece (approx 21 inches long) in half. These are stacked by color or theme on the closet shelf. No extra supplies needed with this method.

You can ask for the cardboard bolts left over from a cutting counter. I have a few pieces of yardage (10 yard bolts) that are left on the bolt. I stack these bolts in an upright clothes hamper for storage. I also store long cutting dies in another one.

https://www.containerstore.com/s/ste...ppins%20hamper

I am 75 years old and like making scrap quilts with my fabric but I am working on downsizing also. When I was cutting fabric for my last strip quilt, I put scraps smaller than 2 1/2" in a bag and when full, it goes in a large zip lock bag. I then drop it off at a thrift store and am told by clerks, those bags are gone the first day they are put on the shelf. So I know they are wanted and used. Anyone doing applique loves small pieces. Gives me a good feeling too.

I will put fairly narrow (5") WOF cuts of fabric back on my stash shelf rather than in the scrap drawer. If they're not WOF cuts, they're more likely to go in the scrap drawer as 'bricks' or strips.

I have any yardage (down to about 1/4 yd) in an old china cabinet to keep dust off it; all potential strings/strips go into a couple bags in the closet for next string quilt; a wire basket of solids only(those aren't small, most FQ size or larger) and then a plastic drawer storage container that holds scraps from 6" sq up to FQ size--but that recently was very depleted making 8 doz. face masks--I figured I used 21 yds total as I was making them for a medical center and they required 4 layers each.

Well, that's really a matter of opinion...
For me, usually, a piece of fabric is a scrap when the project I bought it for (if there is one) is finished. There is no size cut off. I try not to add too much to a scrap bin. I mostly keep reds with the other reds, blacks with the other blacks, Kaffe's with each other and so forth.
Sometimes I add long strings or itty bitty pieces to the scrap bin, but mostly I do this if either I have some scrap projects in mind or I'm trying to clean up some other bin.