What I normally do is cut out the label, throw away the rest of the box, put a rubber band around the cards, and store them with the label on top inside plastic sandwich bags. I put those together in a gallon sized plastic freezer bag. This is simple, cheap, waterproof, and everything needed is already in the house. I can easily see through the bags and know which set is which. It adapts to different sized cards very easily. It doesn't look the best, but I put mine in a desk drawer out of sight and reach of little hands.
Here are some other methods I have come across over the years:
You can use one of the hang on the door shoe organizers inside a closet. Just pop the cards into one of the pockets.
You can put rubber bands on each set and store them in a plastic shoe box.
Repurpose empty baby wipe containers to store each set.
Use plastic reusable food containers (sandwich size) to store the cards.
Buy pencil zipper cases with the clear front when school supplies go on sale and use those.
Find recipe boxes and store them in those.
Find clear cosmetic bags with zippers at the Dollar Tree and use those.
The sheets and pillow cases I buy come with a fabric bag with a snap. If you sew, you could make smaller bags with snaps, zippers, a button, or a drawstring to store your cards.
On Pinterest, I recently saw an idea using plastic index card boxes inside of a plastic shoebox. This looks very neat and organized.
If you have another flash card storage idea that would work in a homeschool setting, please send me a description. Include a picture if you like. I will post it here, and give you credit.
If you are a homeschooling mother, I am praying for your success and well-being. Remember to look to the Lord for answers as you teach your children.