First, make your beet purée! Preheat your oven to 400F. Wrap each beet in foil and roast for 45 minutes, or until the beets can be easily pierced with a fork. When the beets are cool enough to touch, run them under cold water while you slip off the skins. Roughly chop the beets, then add them to a food processor or blender. Blend or process the beets until they make a smooth purée (a couple minutes). Set the purée aside; you should have 1 1/2 cups or so.
Reduce the oven temperature to 350F. Lightly oil an 8 x 8 square baking dish (or an 8 or 9 inch round baking dish).
Whisk together the flour, baking powder and soda, salt, and cocoa. In a separate bowl, whisk together the beet puree, sugar, oil, water, vanilla, and vinegar. Add the wet mixture to the dry ingredients. Mix the ingredients till you have an evenly mixed batter; a few little clumps is fine.
Transfer the batter to your baking dish. Bake for 40-45 minutes, or until the top is firm and set and a toothpick emerges clean (or nearly clean). Allow the cake to cool for 15-20 minutes before removing it from a pan and transferring to a wire cooling rack to cool to room temperature. Cut into pieces and enjoy.