Breast milk will stay tasty in the refrigerator (about 39 degrees Fahrenheit) for about five days, according to recommendations from the Centers for Disease Control and Prevention (although some say up to eight.). For maximum freshness, store the milk in the back part of the main body of the fridge. (The temperature doesn't stay consistently cool near the door.)
If you are traveling and need to store the milk in a cooler bag, that milk will spoil sooner than if it were kept in a refrigerator. Keep the milk in contact with ice packs at all times and feed it to baby within 24 hours.
If you'd like your supply to last longer, consider freezing it: A refrigerator that has a freezer behind separate doors can keep milk safe for up to two weeks. Just make sure to thaw it safely by transferring it to a refrigerator rather than microwaving the frozen milk.