A "museum city" was a municipality, or part of one, devoted to recreating the sights, sounds and smells of a previous era of that municipality.

It has not been revealed, in pro or fan fiction, when these first were created. It seems probable that these were established during Earth's long recovery from World War III and the Post-Atomic Horror. Such museum cities were probably similar in nature to Colonial Williamsburg in the early 21st century.

Alexandria in Egypt was recreated as a museum city by the mid-23rd century. (Star Trek: The Motion Picture novelization)

One of the oldest in North America was known as "Old City San Francisco", near the Telegraph Hill district. (Star Trek III: The Search for Spock novelization)

New York City had a few museum cities recreating different eras. One was in the Harlem district of Manhattan, recreating the ambience of the city's "Harlem Renaissance" period of the 1920s and 30s. (Star Trek: Avenger: "Nearly a Valediction")

By the late 24th century, some parts of Paris were "museum cities", with sidewalks and outdoor food vendors rather than slidewalks and replicators. A newly-minted Captain William Riker and his former commanding officer, Captain Jean-Luc Picard, were in one of these when they happened upon the mysterious bar known as "The Captain's Table." (TNG novel: Tales From the Captain's Table: "Improvisations on the Opal Sea: A Tale of Dubious Credibility")