Generally, new parts are in the bags/boxes they were sent to me in. I keep a record of what I've ordered so I know what I have on hand. I don't keep endless stockpiles of belts, gaskets and things like that, just enough to cover the next service then I'll order some more, oil filters and the like being the exception that I tend to keep a few extra on hand. Where I replace an old part for new that's not totally shot, it goes back into the bag/box the new one came out of, and onto a different shelf. That's for your basic daily items.
When it comes to some bigger metal items I have in long-term storage, like spare diff centres, axles, hubs, things like that, I pick a day without much humidity, slather them with a heavy engine oil, and entomb them a wide plastic roll, basically clingwrap. Pretty comfortable with that long term.