Boiled eggs can be safely kept in the refrigerator for up to 7 days whether they are peeled or unpeeled, as long as they are stored properly. They should be refrigerated within two hours of cooking to prevent bacterial growth. If peeled, it is best to store them in an airtight container with a damp paper towel to keep them moist. Eggs left at room temperature for more than two hours should be discarded. Spoiled eggs may have a sulfuric smell or off taste and should be thrown out. Freezing boiled eggs is generally not recommended because the texture becomes rubbery, especially the whites.