Just look at how I store mine. This was a salsa jar, the perfect size. I crocheted a round top and pulled the string through, a very cute container. I keep mine right on the kitchen counter, along with the linen butcher's twine in a beautiful hand turned (by Mr. B) dispenser.
If you are still a purist, and insist on using authentic baker's twine, go to the source. A bakery supply store, like this one, Country Clean Paper Supplies. Here you can purchase a two pound cone, that 's 3400 yards of 4 ply twine in at least 5 twist colors, for $8.95-$10.50. I may do this in order to get the black/ white twine I need. Now go get some decent twine...it's hot, hot, hot!