You can spend $70 or more for a stripping basket or you can make one for just $5 by using a stool from Ikea and a few plastic bolts used to attach a toilet seat. You can buy them at hardware stores.
You will need to remove with a file half of the thread from the plastic bolts so the fly line will not "catch" on the thread. Cut two channels and use a wading belt to attach the basket around your waist. A few holes drilled into the bottom will help drain the water when it's raining. Or make no holes if you wish to keep some water into the basket to lubricate the line (useful when saltwater fishing).
Using plastic threaded bolts instead of fixed bolts allows you to remove them for transportation and use the entire basket for safely storing fly-boxes inside.