The only people who can walk around armed are MPs or people on their way to/from authorized training.

Uh huh... I've carried a Mk-14 clear across base before. Nobody stopped or said shit...

If you live in barracks on-base and own a personal gun, if often has to be stored in the base’s firearm storage.

No... your units weapon storage... and that only because you can't bring a gun safe into the barracks. Keep in mind that Barrack != on post housing... You can have a gun safe and many guns in your on-post non-barracks housing.

Often even a paintball gun has to be stored in the base armory.

No in my experience on 4 different posts.

Even if you have a concealed carry permit for the state the base is in

Cause Federal land doesn't need to recognize local laws. It's up to the post commander what they want to do.