How To Store. Uncooked asparagus will stay fresh for three to four days in the refrigerator. The secret is to keep the vegetable cool and damp. Store spears upright in a container with the stems wading in an inch of water, then cover loosely with a plastic bag.ou can peel stalks up to a day before cooking and store them wrapped in damp paper towels in the vegetable bin of the refrigerator. Conventional asparagus wisdom says snapping the spear breaks away the tough, fibrous end.