Ordinary household storage is good enough for small quantities. Just leave it in the box and put it in a drawer or on a shelf and you should be fine for years. If you're worried about someone getting into your ammo that shouldn't, store it in the safe or even just a lockbox can make it more secure.
Now if you have larger amounts to store you may need something a bit better. As a former dealer and a current pack rat I tend to store a lot of ammo. Still have several crates of military surplus put back. Stored on a raised platform in the basement they should be good for more years until I shoot, sell or gift them. Most of my shooting ammo is stored in GI ammo cans, with good seals they will last a long, long time. The cans have also been money savers in a way, due to a drain pipe collapse I had my basement flood a while ago. Quite a few cans were underwater for a time and I didn't lose a single box of ammo because the cans stayed dry inside (also why the crated stuff sits on a raised platform). While I may not experience another such flood I prefer knowing that things are as safe from flooding as they can be.