Depending on how many string envelopes you want to store, a large KLEENEX box has sufficient cross section to hold many, many sets of strings - and if you want fancier you can use the box as a starting inside mould for making a more durable one out of any material.
Because I have had strings for violins, violas and cellos (for more than 70 years) I have a larger box: the central drawer in a 3-drawer plastic chest (10wx12dx7h, inches). (The other two drawers are filled with other string-playing paraphernalia.) I have always saved my "gently used" strings, but fortunately just as the string drawer was getting too full last year a young man in New Jersey posted here on violinist.com soliciting used strings for donation to youth ensembles (worldwide) and I sent him the strings I probably would never again find use for. My violin string envelopes still fill the full depth of one side of the drawer; the other side now has plenty of room for more viola and cello strings that it will probably never see.
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I use a nice wood box that was originally made and carved in India. My wife subtly colorized it artistically and gave it to me decades ago. It has a neat appearance, and it sits on my violin table where I place my violin case while practicing.