Boil beets 25 to 35 minutes, depending on size, until tender; set aside to cool. Meanwhile, cut turnips or radishes in half and slice into thin half-moon shapes. Slice red onion to an equal thickness. In a medium bowl, combine turnips, red onion, olive oil, red wine vinegar, sugar, salt, and pepper; let the mixture “pickle” at least 2 hours, or overnight.
Peel the cooled beets, then cut in half and slice thinly, keeping all vegetable slices about the same thickness. Add beet slices to the turnip mixture. Toss well, then chill. Garnish with chopped tarragon before serving.