There are three main conditions to store your veggies: cool and dry, cold and dry, and cold and moist. Basements are cool and dry and fridges are cold and dry. To achieve cold and moist conditions, certain modifications need to be made. The produce or crisper drawer keeps humidity decently well and is better than the open fridge shelves for many veggies, but is still not ideal. Putting vegetables in slightly open or perforated bags in your crisper drawer is as close to ideal as most can get at home. This holds in humidity but allows airflow to prevent mold and decay. Keeping things in bags on fridge shelves can work for periods as well. Garages that don’t freeze but stay cool can be good places for storing cold and dry produce as well.
Cool and Dry (Basement):Winter Squash, Pumpkins, Onions
Cold, Dark, and Dry (Fridge):Potatoes (ventilated container/perforated plastic bag), Garlic
Cold and Moist (Fridge, plastic bag and/or crisper draw): Cabbage, Kohlrabi, Beets, Carrots, Parsnips, Potatoes, Radish, Turnip, Kale, Chard and Leeks
Avoiding Limp Vegetables: (1) Remove tops: when storing crops such as carrots, beets, turnips and radishes, take the time to remove the tops (if they have them). If left on the tops draw moisture out of the root, leaving it limp and squishy. Simply cut off the greens, leaving a 1/4 inch so as not to cut into the root. (2) Re-hydrating: if your vegetable does become limp, often it can be salvaged as long as it hasn’t molded. Simply re-hydrate it in a bowl of water. Vegetables with a thicker skin such a beets and potatoes can be skinned before placing in water, others such as carrots or celery can simply be cut off at the end and placed cut end down in a cup of water. (3) Keep in Bags: greens such as kale and chard, and topped root vegetables such as carrots and beets, are best kept in the refrigerator in a bag. The bag should either have perforations or be left open a little at the top for airflow.
Check out this comprehensive guide from Cornell. This guide provides you with details about temperature, humidity, and length of time the specific vegetable can be kept.