Unopened melons can be kept on the kitchen counter. There is absolutely no need to place them in the fridge. Studies have proved that the levels of antioxidants that whole melons contain stay the same if the fruit is stored in room temperature. However, low temperatures will make the food lose its beta-carotene content. If the melon has been opened and sliced, you can safely keep the leftovers in the cooler. Remember to cover the food.