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, 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.