I used to store my fabrics, in folded stacks on inexpensive Staples 28" wide x 70" high bookcases. Because the thin shelves bowed from the weight, I bought 3/4" melemine shelving blanks at Home Depot and cut them to fit my bookcases. No more bow and 6 or 7 shelves per case! Stacked by color families and type of fabric (dress, home dec, quilting in color ways). I just closed my miniblinds in that room when I wasn't working. I suppose you could also put spring tension rods in the top of each case and hang a sewn to fit cover down over the shelves.
But THEN, we had to renovate the house. My sewing room dimensions didn't change... but when I redid my sew room, I purchased three 24" wide x 16" deep RSI/Estate cabinets from Lowes - the models with the nice smooth finish inside and out. I refolded my fabric to fit the shelves. Now I close the doors.
On plastic: I used to also have fabrics in plastic bins. Silver fish are a huge problem in florida, so plastic it HAS to be. But I took care to keep the A/C on... and would open up the tubs to air the fabric out on occasion. :lol: