Fruits and Vegetables can be tricky to store because some fruits and vegetables are incompatible when stored together. Some fruits emit ethylene gas which can cause vegetables to spoil prematurely. It is recommended that you keep these "gas releasers" out of the fridge: avocados, bananas, nectarines, peaches, pears, plums, and tomatoes.
You can refrigerate apples, apricots, canteloupes, figs, and honeydew, but keep them out of the vegetable bin/crisper where you may be storing ethelyne-sensitive vegetables.
Speaking of the vegetable bin, most standard fridges have a vegetable crisper designed to keep Fruits and Vegetables firm and fresher for longer, and sometimes come with moisture and temperature controls. This may be a good place to keep your gas-sensitive vegetables, as the area is sealed off from the rest of the fridge.