I use 2 types of abrasive:
Cloth backed bought on a roll, with the roll in a box. Push a length of plastic pipe through the side of one box, through the centre of the plastic roll and out the other side of the box. Do the same with the other boxes so that the grit is in the right order. Then run a piece of string through the pipe and hang both ends from nails.
Velcro back discs, ready cut. These come individual cardboard boxes for each grit. I have a long wooden box which is wide enough for a disc. I've glued plywood separators along the box, one for each grit. I usually keep about 5-10 of each grit in this box to make them easily accessible. The individual cardboard boxes are stored elsewhere.
For the discs I'm currently using I have a length of velcro glued and screwed to the roof directly above the lathe, within arms length. When I want to change grit I simply remove the current one from the arbour, replace it in the correct location on the roof velcro and get the next piece that I want.
I also have a length of loop velcro glued under a shelf - I use this for storing my arbours neatly out of the way