Part of the space was dedicated as a market in 1791, during the colonial period. Wikipedia says it began as a Native American trading post before European colonization, with no date given. The oldest surviving structures date to about 1813.
The Works Progress Administration made major renovations in the 1930s. The market spans six blocks, from Café du Monde to the flea market stalls across from the New Orleans Mint.
Wikipedia calls it "the oldest of its kind in the United States."