Whole wheat contains perishables, such as wheat germ, and I keep my whole wheat flour in the freezer. (I keep wheat germ in the fridge.) Whole wheat bread, however, is kept on the shelves. (So is whole wheat flour for that matter, but it is unopened, as is wheat germ.) I’ve been keeping my whole wheat bread in a bread box. I use it up within a week anyway. Is that proper?
Are you asking about wheat or bread? Actual whole wheat is stored in bins at ambient outdoor temperature for months on end.
If you use your bread up within a week it doesn’t really have time to go bad, but if you are concerned you can always keep it in the freezer like I do and it will remain fresh for weeks or even months.
Whole grain, whole wheat or enriched flour white bread all tend to go moldy after 7 or 10 days unless you refrigerate them. I don’t really like refrigerating my bread but do anymore since I really can’t finish a loaf in less than 3 weeks by myself.
Unground kernels of wheat, ‘whole’ in the sense of intact, keep pretty well at room temperature. That’s the whole point of a seed.
Whole wheat flour is ground up and the oils are now exposed, so they can go rancid-- this just gives it a bit of an off flavor, but you might not even notice. It’s not a bad idea to keep it frozen or refrigerated. It’s not a quick process, so flour can sit out pretty well for a while – possibly months.
Whole wheat bread is moist and a good spot for mold. How fast it goes moldy depends a lot on the conditions and what kind of bread exactly – could be a couple days, could be a week or two, or it might just dry out before it ever molds. If your bread regularly gets moldy before you eat it, you might want to keep it in the fridge, but if mold isn’t a problem, there’s no need.