1. Preheat oven to 350°. Trim beet stems to 1 inch; gently wash beets, and place in an 8-inch square pan. Add 1/4 cup water, and cover with aluminum foil. Bake 1 hour and 10 minutes or until tender. Uncover and cool completely (about 30 minutes).
2. Cut a 1/4-inch-thick slice from each end of grapefruit using a sharp, thin-bladed knife. Place, flat ends down, on a cutting board, and remove peel in strips, cutting from top to bottom following the curvature of fruit. Remove any remaining bitter white pith. Holding peeled grapefruit over a bowl, slice between membranes, and gently remove whole segments. Reserve 1/4 cup juice.
3. Whisk together olive oil, next 4 ingredients, and reserved 1/4 cup grapefruit juice in a bowl.
4. Peel beets, and slice into wedges. Arrange arugula on a large platter; top with grapefruit and beets. Drizzle with vinaigrette; sprinkle with pistachios. Add salt and pepper to taste.