While Whole melons should be left out at room temperature in order to allow them to ripen, they should be stored in the refrigerator once sliced. Food Network comments on this citing research that the United States Department of Agriculture "found that storage at room temp might even help keep the antioxidants more intact."

Get more health and nutrition benefits from your melon by leaving it out for a little while on the counter to ripen, and then store it in the fridge for three to four days once you've sliced it. However, despite the benefits of leaving the melon at room temperature, you may not want to leave chunks of watermelon or cantaloupe out in the open for obvious hygienic reasons.

However, storing the melon is a bit more complicated than stashing them inside plastic containers in the fridge. In case you want to save some melon for later, don’t remove the seeds, to extend its shelf life. Cover the melon slices in a plastic wrap and for cubed melon, lay a paper towel inside the plastic container you store it in, to help prevent moisture build-up.