Regarding potatoes, ideally you should keep them out of the refrigerator. They should be stored in a cool, dark, well ventilated place for use within 3-5 weeks. Cold temperatures can turn the starch in potatoes to sugar and may darken the flesh when the potatoes are cooked. But potatoes stored out of refrigeration often sit around for long periods–especially if you buy large quantities–in temperatures that are often too warm. They get "eyes" and turn soft with wilted skins. If you don’t have a place to store potatoes out of the refrigerator, buy small quantities that will be used within a week or two. Take them out of the refrigerator a few hours or more before using them. This helps to return the starch content to normal.
In general, the ideal storage for whole onions is hanging in a well ventilated dry, cool place. Onions should not be kept near potatoes. The onions absorb the moisture of the potatoes causing them to sprout and rot! Although onions can be stored in the refrigerator, other foods may absorb their odor. They can be tightly wrapped in foil to help prevent this. Cut up onions can be stored in the refrigerator in an air-tight container for several days.