Wood products contain acid that will gradually eat away at fabrics. This includes regular paper, including freezer paper, tissue paper, etc. -- all wood products. Paper that has been treated to remove the acid is sold as "acid free" or "archival quality" paper, but it is very expensive.
The contact paper idea sounds good. You can also line the shelves and drawers with fabric. Inexpensive, washed white sheets can be cut up to act as liners, providing a barrier between the wood and your stored fabrics. I would maybe spray-baste the white sheeting to the shelves and drawers.
As someone else mentioned, polyurethane is safe. One way to avoid using liners is to cover the wood with a coating of polyurethane. (I've never done that, so don't know easy or hard it is!)