Preserving whole potatoes
The ideal storage for potatoes is approximately 42 to 50 F /  5.6 - 10.0 C in a dark space, with good humidity.  This is historically what root cellars were for.  With proper storage you should be able to get them from one harvest to the next&mdash;but ideal storage is rare at home.  Nonetheless, at least a few weeks is more than reasonable.

Burying them in clean sand can be effective.

For more information than you ever wanted to know, see this article from Options for Storing Potatoes at home (PDF) from the University of Idaho extension.