If it's a cardboard box, there's no fully sealed state, as cardboard is not impervious to moisture and such, even if the box is a case or a flat. Cardboard is cardboard. In fact, in a very high humidity environment, the cardboard boxes would be a liability in trapping/retaining moisture. The MTM boxes or a Magnum/Plano polymer ammo container with the rubber seal are all great for storage, like what you pictured. Princess Auto had the 50 cal boxes on sale for $7.99 each not that long ago (about a month ago). I believe they have a current sale on the metal cans.
Others keep ammo in their original boxes in ammo cans or in a cabinet or safe in an environment which is dark, cool and dry. Ammo is generally pretty resilient and there are folks who report that they successfully shoot 60-80 year old ammo (way before the advent of common place air conditioning) with no problems. Some folks take all their ammo out of the boxes and put them in the MTM cases or the polymer or metal ammo cans.
The biggest thing is keeping it away from direct sunlight, quick changes in temperature, and obviously moisture. Many people who keep ammo in a cabinet/safe have some sort of desiccant tin/Golden Rod solution handy, or control the room the ammo is stored with some sort of dehumidifier. Even if it is not that elaborate, keeping it dry and cool goes a long way in and of itself.