When you pick a fruit or dig a root, it’s still alive and respiring. Reproductive parts, like seeds and nuts are also very much alive, and just sleeping until it’s time to grow. Live storage, when you can manage it and for appropriate foods, is the storage method which retains the most nutrition and potentially costs the least.
If you live north of the Mason Dixon line, or in a similar climate, you are in luck: you can build and use a root cellar or similar techniques. The best book on the subject to date that I have read is Mike and Nancy Bubel’s Root Cellaring. One weakness of the book is that they give very specific harvest and planting dates which may not apply in your area, but they cover numerous storage crops and methods, best temperatures and likely storage length.
For those of us in warmer zones, root cellars simply won’t get cold enough. For foods needing the coldest storage (32-40F), you may very well be able to “store” those crops right on their living roots in your garden. While few plants will grow at those temperatures, you start winter crops like cabbage outdoors in the late summer and they grow heads in the cool temperatures of fall, then sleep all winter until you are ready to harvest and eat one. For crops which like slightly warmer temperatures (40-50F), we are out of luck for non-electrical means, but a dedicated refrigerator or Cool Bot may be cost effective depending on how much food you are storing.
The plants which like or tolerate the warmest temperatures, like winter squash, sweet potatoes, pecans and other nuts in the shell and seeds in the shell, a slightly cool basement or room will work fine. I keep a storage room in my basement which stays about 60-65F in the winter. Here are some of my recent squash harvest stashed on old towels in plastic tubs, but any storage which allows them to breathe, has enough cushion to prevent bruises, and mostly don’t touch each other will work.