Scrub beets; trim stems to 1/2 inch. Place beets in a medium saucepan; add water to cover by 1 inch. Bring to a boil over high heat. Reduce heat to medium high; cook at a gentle boil until beets are tender when pierced with a small, sharp knife, 30 to 40 minutes. Remove from heat, drain, and set beets aside until cool enough to handle.
Meanwhile, grate horseradish on small holes of a box grater. Place in a small bowl; stir in vinegar, salt, and sugar.
Peel beets; grate on small holes of box grater. Add to horseradish mixture; stir well to combine. Keep covered to preserve flavor, and refrigerate until needed.