The 2009 video The Story of Stuff makes several rather incredible claims. For example:

Where I live, in the United States, we have less than 4% of our original forests left.

My gut feeling tells me to be skeptical about this number. Is it correct?

Yes, this figure is consistent with estimates from 20 years ago.

The 1995 paper Endangered Ecosystems of the United States:
A Preliminary Assessment of Loss and Degradation collates some relevant estimates from the literature in Appendix A.

50 United States

85% of original primary (virgin) forest destroyed by late 1980's (Postel and Ryan 1991).

90% loss of ancient (old-growth) forests (World Resources Institute 1992).

[...]

48 Conterminous States

ca. 95-98% of virgin forests destroyed by 1990 (estimated from map in Findley 1990 and
commonly estimated by other authors, e.g., Postel and Ryan 1991).

99% loss of primary (virgin) eastern deciduous forest (Allen and Jackson 1992).

It goes on to break the USA down into smaller regions, and cites consistent statistics for those - e.g.:

>99% loss of virgin or old-growth forests in New Hampshire (D. D. Sperduto, New
Hampshire Department of Resources and Economic Development, Natural Heritage
Inventory, Concord, N.H., personal communication).

This shows it is more than one or two papers making these nationwide estimates - they are shored up by several ecologists who have reached similar conclusions in different regions.