What edible fats or oils store well without refrigeration?

All oils will store well for about 6 months. Unfortunately most long term fats will go rancid. Anything canned will last longer.

I personally keep an extra bottle of olive oil in the freezer and rotate it when I buy a new bottle. Keeps it very fresh, and in the long term it will keep indefinitely.

It doesn't take up much space either.

I keep oils in my cabinet next to the stove.

Never had issues, while my olive oil is always empty within 2 months the sunflower oil I am still using since 15 years and never had to buy a new one