I think it's OK to store in plastic, as long as eventually it comes out and goes into a quilt where it can breathe a long sign of relief!!! :-)

My fabric is in plastic bins in a cabinet but I take it out periodically to pet it.

Thanks for the information

They often have inks and imprints that will leech into the fabrics, too.

I was told not to store my quilts in plastic bags, but pillow slips. The wadding was the problem not the fabric.

It also helps to toss in a desiccant packet - those silica gel thingies you find packed in with electronic equipment or in pill bottles. They keep the humidity down in a sealed plastic tote.

When I give a quilt I try to give it in a pillow case. I used to buy them cheap at the linen discount stores but now they are gone. Some times I have enough fabric left to make a matching pillow case. On the other hand I kept the quilt my mother made for me when I was little (and I'm in my 70's) in plastic for a long long time, and no problem.

Fabric should not be stored in plastic containers. Fabric needs to breath. Plastic boxes outgasses formaldhyde. The fabric therefore can become toxic from this outgassing. Fabric pieces can be stored in a baggie while you are working on a project to keep pieces sorted. But long term is not recommend since the fabric isn't able to breath.
I'm sure this is a topic with a lot of opinions.

Polypropylene, the most common plastic used for storage tubs of the type used for fabrics (it's marked with a 5 and the letters PP), contains no formaldehyde. It is, in fact, also used for drinking straws and yogurt containers. I find no reference anywhere of plastics containing formaldehyde in today's production. Plastics may or may not outgas chemicals, but formaldehyde is not one of them.

On the other hand, fabrics are coated with the stuff as part of the manufacturing process. It wouldn't matter where it is stored, if it's not prewashed it is outgassing formaldehyde.

You also forgot about storing them in cedar chest which also is a no-no.