Find the depth of the frost line in your area. Your local building authority---whoever issues building permits in your area---will know the number, but you can also speak with a local contractor; they have to deal with frost line issues all the time. The frost line is the maximum depth where groundwater will freeze.
Dig a hole in a well-drained location convenient to your kitchen. If you've got an unheated basement, this phase of the work can be done for you, because you can use a corner of your basement. Otherwise, an excavation in a hillside makes for easy access and good drainage; but even a hole in flat ground can work. Make sure your hole extends at least a foot or two beneath the frost line.
Line the hole with concrete block or other moisture-tolerant and vermin-proof material, and seal off the entrance with a door. The airspace between the door and your food storage will help serve as an insulator.
Place an incandescent light bulb in your root cellar. Should the temperature remain extremely low for a long period and threaten to freeze your produce, you can turn the light on to prevent freezing. This should only be necessary in extreme circumstances.