I storage binding by wrapping it around empty cardboard salt containers. To work with it, I roughly punch out a hole in either end. I take a plastic hanger and cut one side through near to the top and then 'thread' the unit on to it, enabling me to hang it up nearby my machine (usually from my floor lamp). Works for me.
[I don't worry about the fabric of the binding being next to the cardboard---but if I make binding ahead of time, know I will be storing extra or have a light colored fabric, I may cover the cardboard with something first.]