There are a number of alternatives for storage of your family's paper-based heritage. First, if you must store these materials in boxes, invest in archival-safe storage boxes. These are available from a number of companies. Local photographic and some office supply stores may carry these items. If not, contact companies on the Internet.
Binders can provide an excellent storage facility.
Make sure they are archival quality, or at least insure that every document is encased in a sheet protector. Binders are portable and easily read. Using top-loading polypropylene sheet protectors, the documents can be handled without fear of deterioration. A nice feature of using this storage method is that documents of different sizes can be made more uniform for storage and display when inserted into sheet protectors. Tape, however, deteriorates over time, and the adhesive will definitely leave a residue on the document.