Mom and dad's cellar never goes below freezing, even when the temps. go below zero outside. Thick walls and insulation up front (even the door) and dirt piled up in the back and most of the sides.
Dad made some big wood boxes, about 4 feet long by 2.5 feed wide by 2.5 feet deep and used them to store potatoes and apples.
Just like apples, some types of potatoes store better/longer than others.
No lights in the cellar except when they went in to get some of the canned vegetables from the garden.
Yup, just pick up the potatoes, one by one, and rub your fingers over the surface to bust the "eyes" off them.
In the spring, when you get ready to plant, cut them up so that there is an eye or two on each piece you plant. The eyes will be either the roots or the stalk/plant (I can't remember which).
I've dug potatoes in late August (when we took in the garden at home) that had the piece of potato the plant grew from still on the roots and looking like it had just been planted in the dirt a few days before you dug it up. Most of the time there's nothing left of it, but sometimes it's still there.