Preheat the oven to 375°. Arrange the beets in a baking dish and fill with 1/4 inch of water. Cover with foil and bake for 45 minutes, or until tender. Let cool, then peel the beets and cut them into 1/2-inch-thick wedges.
In a very large pot of boiling salted water, cook the beans over high heat until crisp-tender, about 5 minutes. Drain and refresh under cold water. Drain well and pat dry.
In a small bowl, whisk the olive oil with the vinegar until blended. In a large bowl, toss the beans with 1/2 cup of the vinaigrette and transfer to a platter. In the same large bowl, gently toss the beets and chives with the remaining 1/4 cup of vinaigrette and spoon over the beans. Season with salt, sprinkle the feta on top and serve.
The recipe can be prepared through Step 2. Refrigerate the beans and beets separately overnight. Let return to room temperature before proceeding.