Whisk ¼ cup oil, white wine vinegar, and sugar in a large bowl to blend; season with salt and pepper.
Add kale and shallots; mix until completely coated. Cover and chill until kale is tender, at least 3 hours.
Meanwhile, preheat oven to 375°.
Peel beets and cut beets into small cubes. Arrange beets in a small baking dish and drizzle with a little oil. Season with salt and turn beets to coat. Bake beets until tender when pierced with a thin knife, about 30 minutes.
Cook barley in a large pot of boiling salted water until just tender, about 45 minutes. Drain barley and spread out on a rimmed baking sheet; let cool completely.
Add beets, barley, and feta to kale. Drizzle salad with remaining 2 tablespoons oil and 2 tablespoons rice vinegar; fold gently to combine. Season to taste with pepper and more rice vinegar, if desired.