This is too important. The paper helps cushion the material. Sharp folds and creases actually break the textile fibers and cause more damage. Steel safety pins and paper clips will rust over time, regardless of how carefully you think you are controlling humidity. They will leave rust stains that probably can never be removed.
Plastic bags are great for short-term storage of materials from the cleaners.
However, remember that plastic is a petroleum-based product. Over time plastic breaks down. It gives off fumes and chemicals that can discolor and destroy many fabrics. Never store precious fabrics with polyester materials. Your grandmother's wedding dress should not be stored forever beside your husband's favorite leisure suit from the 1970s.