There are 2 or 3 parts to your question. You've got your ammo in a cardboard shebox right now. That ammuntion is insecure and subject to dust, humidity and handling. A metal (or plastic) container will keep unwanted fingers out of your "stash", protect the ammo itself from the interior elements, and is strong enough to carry around.
Why you wouldn't want ammo cans must be something personal. They are dust, humidity and (sort of) fire safe. They have strong handles that fold. A clever fellow can make them lockable. And they are cheap. I have enough of them that I forget what is in each one, so I made cereal box cardboard tags that are tie-wrapped to the handle.
As an alternative, check out mechanic's toolboxes, stationary store file boxes or small Rubbermaid tubs. Your choice.