I'm wondering if there is a way of keeping vegetables and fruits fresh without using a refrigerator.
basically food like potatoes, tomatoes, bananas ...
My answer is based on the assumption that you are looking for an alternative refrigerator which does not require electricity.
Use can use Pot-in-pot refrigerator.
A pot-in-pot refrigerator is an evaporative cooling refrigeration device which does not use electricity. It uses a porous outer earthenware pot, lined with wet sand, contains an inner pot (which can be glazed to prevent penetration by the liquid) within which the food is placed - the evaporation of the outer liquid draws heat from the inner pot.
The device can be used to cool any substance.
This simple technology requires only a flow of relatively dry air and a source of water.
Alternatively, you can put the vegetables in steel utensil and cover the whole utensil with clean wet cloth.
However, this alternate solution does not cool better than Pot-in-pot refrigerator.
Since most food goes bad because of bacterial growth, exposing the food to UV light is often used as an alternative to thermal treatments (freezing/cooking).
Also, putting meat in an enclosed space that has salt blocks for dry-aging works. Rather than salting the meat directly, which changes the flavor, the salt blocks affect the air in an enclosed space and bacteria growth is hindered allowing the meat to age with minimal bacterial decomposition while allowing the natural enzymes to break down and tenderize the meat.
(plenty of google references for this technique)
Before refrigerators a piece of marble or another material object that feels cold* I can not give sources for proof that this works or get my head round why this works.
*yes I know there is a difference between feeling cold and being cold, see https://www.youtube.com/watch?v=vqDbMEdLiCs for more info
If you want long-term preservation, then some traditional methods work well:
Dehydration - Dried apple rings, apricots and banana chips are some examples.
Osmosis - Add enough sugar or salt and spoilage organisms grow slowly. Jams, jellies, marmalades and other preserves rely on this principle.
Blanching - Quick-cooking just enough to delay enzymatic and bacterial degradation.
Look online (e.g. Mother Earth News) for more ideas.