Harvest to Table gives these tips for storing turnips: Don't wash turnips before storing. Store turnips in a cold moist place as near to freezing as possible without actual freezing, 32°-40°F (0°-4°C) and 95 percent relative humidity. Store turnips wrapped in a moist cloth or paper towel in a perforated plastic bag in the vegetable crisper drawer of the refrigerator. Turnips will keep in the refrigerator for 4 to 5 months.