What's the best place and way to store potatoes so they keep fresh longest?
Best thing to do is to keep them out of the light in an cool dark dry place. I usually put them in a hessian bag to try and keep the moisture away.
Don't store them in the fridge or anywhere that gets direct sunlight.
Potatoes should be stored at temperatures between 7 and 9 degrees C (45 to 50 degrees Fahrenheit). The starches in the potato will start to break down into sugars at temperatures below 7 degrees making for a darker, more bitter tasting result after frying or roasting.
Many people say that you shouldn't store potatoes in the fridge. If, like me, you have a compartment in your fridge that goes above 7 degrees, I believe it is reasonable to keep them there despite common misgivings.
Air circulation is important according to one internet source to "remove the products of respiration". A cool, well ventilated cellar would be the ideal place to store them.
The first article I linked to seems to suggest that you can store potatoes at lower temperatures for a few months.
I recently bought this jar to hold potatoes. It has holes in the bottom to allow a little bit of circulation. So far I am very happy with it.