Perhaps it's your mother's wedding dress, your child's baby clothes, your father's military uniform. I have pillowcases with my Grandmother Weatherly's handmade lace edging, and a hand-crocheted baby set made by my Aunt Mary Allen. All of these are important parts of your family tradition , things that add special significance and understanding to your understanding of their lives. These are tangible evidence of your ancestors' existence and their handwork. It is your responsibility to preserve these items and help carry on the tradition.
Many dry cleaners have acid-free boxes and tissue for sale.
Be sure to specifically ask whether the materials are acid-free. If the person doesn't know or isn't sure, go somewhere else. This is too important. The paper helps cushion the material. Sharp folds and creases actually break the textile fibers and cause more damage.