Potatoes are best stored:
In the dark. Potatoes turn green in the light due to the production of chlorophyll. Light also induces the production of the harmfull glycoalkaloid solanine that develops along with the chlorophyll.
At between 42 and 55 degrees farenheit (6 and 13 degrees Celsius), 45 degrees (7 degrees Celsius) is apparently best. Above this the warmer the temperature the faster they will sprout.
Humid ventilated environment. Don't keep them sealed in a plastic bag. Special potato sacks help keep them well ventilated.
At temperatures below 42 degrees (such as in your fridge) some of the starch in the potato gradually breaks down into glucose. This happens quite slowly, I have never noticed any effect from keeping potatoes in the fridge for a week or two. The conversion to glucose only really becomes an issue when you are deep frying your potatoes to make chips, the sugar can result in dark coloured potato chips. If you bake or boil your potatoes these effects are not noticeable. Refridgeration is only raelly really a problem for long-term storage.
There isn't anywhere in my flat (apartment) cool enough to store potatoes without them sprouting so I always keep them in the fridge. I eat a lot of potatoes so they rarely last more than a week.
Ideally you would have an unheated cellar or outroom in your house or you could keep them in a spare vegetable fridge set at a warmer temperature.
This is a good source with even more information: Options for Storing Potatoes at Home
