Store the whole dragon fruit at room temperature. Ripe dragon fruit should be stored in a sealed plastic bag in the refrigerator for up to 1 week. This technique prevents it from picking up smells from other foods. If underripe, you can accelerate the ripening process by placing it in a paper bag with an apple or banana. The ethylene gas emitted from those fruits accelerates ripening. For cut pieces, store in an airtight container for about 2 to 5 days, but discard when the flesh turns brown or mushy.