We have harvested our horseradish roots and are storing some to re-plant in the spring and preserving some to eat over the winter.
To store horseradish for spring planting: Cover root in sand and store at 32-40 degrees or keep in the back of your refrigerator (but be sure it doesn't get wet).
To preserve horseradish for eating: Grate the washed root finely. Process in a food processor with vinegar and salt (1 Tbsp:1 tsp). The food processing stage really brings out the roots heat (just grating it won't get you there). Store in a sterilized jar in the refrigerator.