I store them in the tube they came in. They are cleaned and lubed after each use with the Ridgid Extreme cutting oil we use to cut with before they go back in the tube. Then they are all stored in plastic organizer drawers by caliber. .17, 22, 243/6mm, 257, 6.5/264, 7mm/284, and so on up through 50. Same for all the bushings. My shop is heated so never felt the need to check on them while stored. In 6 years they continue to be razor sharp and look brand new.
I have one reamer, that when I got it back from the guy who used it, had been dipped in a soft plastic like material. This soft coating will slide off and on the reamer easily, stays oily, and will still fit into the original plastic tube. This appears to me to be a better method. I was curious if anyone used this stuff.