It appears to have been built in 1861, but that date is somewhat uncertain, due partly to the fact that it is one of the oldest structures in Amador City. (The 1851 date you may have noticed painted on the front of the building is a fairly recent, and inaccurate, addition. As in most gold rush towns, fire destroyed many of the earliest structures. Amador City suffered particularly serious blazes in 1876 and 1878, each of which burned most of the downtowns wooden structures.
The museum shows and example of a one room school house of the early days. If a settlement had at least 5 school aged children.
The building is named for William Fleehart. Born in Ohio in 1828, he is known to have been in Amador County in 1856, and to have had interests in various mines and mercantile activities. It has not been established that he actually constructed the building, but ran what was probably the first business operation from it: Gardner & Fleehart General merchandise. Receipts in the Amador County Archives indicate that they dealt in hardware and seed, acted as a telegraph office, and were Wells Fargo agents shipping bullion at least once a month.
Gardner and Fleehart declared bankruptcy in 1869. The store ended up in the hands of the Kirkland brothers, who continued to operate it as a general store and a Wells Fargo agency. William Fleehart died in 1890. Kirklands sold the store in 1915, after which it passed through several hands and incarnations, including one as a Gold Rush Trading Post and Museum.
Information, photographs courtesy of the Amador County Archives, The Historical Marker Database, and the Chronicling America Database