The game can be played without a Craft Bag. I did for a year or so until they created it. Although, this was back when there were a lot less Style Mats, maybe 15 or 20? So, it was easier. I do remember destroying stacks of style mats whenever I got to 200. Happened quite often.
With the Free ESO Plus periods happening 2 or 3 times a year, you can off load your hoarde periodically.