Root cellar. I don't know how these work in warmer parts of the country, but we've always used these to store harvest and perishables. At our cabin, our main root cellar was below the cabin floor, dug straight down probably 10 feet? (I don't know exactly, I remember my parents would climb down into it and their head would be significantly below the floor). It was maybe 3x3 or 4x4, with shelves along the side, and we stored perishables there. We had an additional root cellar outside that was dug into a hillside where we kept root veggies.
My parents currently live in a strawbale, solar powered house. They have freezers but no refrigerator. They use an "icebox" for a fridge, built like an under the counter fridge with lots of insulation, and they change out ice packs from the freezer twice a day. They also have a root cellar under the first floor, where they keep root veggies. The carrots they usually store in sand or dirt. They freeze, pickle or can the more perishable veggies like zucchini, broccoli, brussel sprouts, kale, etc.
This is all in Alaska, though, so I'm not sure what that looks like in warmer climes.
Here's an article about
root cellars.