Chop off the stem and greens, leaving about 1-inch of stem attached to whole, unwashed beets. Store the beets in the crisper drawer of your refrigerator. They usually stay fresh for several weeks, but will start to shrivel and soften over time. Firmness is a good indicator of how fresh the beets are. Beet greens only stay fresh for 3 to 5 days; it's best to use the greens as soon as possible. They can be stored in a dry produce bag in the refrigerator.