To store ripe papayas, place them in a plastic bag and put them in the refrigerator. If they are left at room temperature they will continue to ripen and become overripe.
They will keep up to a week in the fridge but they are at their best if eaten within a day or two.
To freeze papaya, remove the seeds and cut into cubes or slices. Place them on a cookie sheet and freeze. Once they are frozen, put the pieces into freezer bags or containers. (The cookie sheet method prevents the pieces from freezing together into one solid mass.)