Watermelons: Look for symmetrical unblemished watermelons, without flat sides, that have a creamy yellow spot on the bottom indicating ripeness. At 92% water, this fruit should feel heavy when you heft it. Precut melon flesh should be dense, firm and appear moist. Store uncut melons in the refrigerator for up to a week or keep in a cool, dark spot. Cover the cut surface of melon with plastic wrap and refrigerate.