Currently I store all my razors in their coffins. I don't even use any oil on them after shaving but I live in a dry climate so it seems to be okay. I am going to be buying a wooden storage box soon though.

After shaving I'll run them under the hottest tap water. Dry them on the towel and strop them 10x on linen. So far no problems. I also must note that I never get the razors wet past the cutting edge either. Water in the pivot area is a sure fire way to get rust.