Potatoes are best stored in a cool (40-45°), dark place. Darkness will prevent greening and coolness will prevent sprouting.
Onions like cool temps (35-40°) and low humidity.
Winter squash like to be stored at about 50-60° and need good air circulation and 50-70% humidity.
Potatoes, onions, and squash will store well for a couple of weeks even in conditions that are less than ideal—under your kitchen sink, for example.
Carrots, beets, cabbage, parsnips and rutabagas: These store best in a plastic bag, or other closed container, as close to 33°F as possible. Cold temps prevent sprouting; closed container maintains high humidity and prevents rubber carrot syndrome.