It is possible during the mid-1950’s to have an integrated orphanage. However, the specific orphanage within The Queen’s Gambit would be highly unlikely to have existed, largely because of how specific of a location it is placed in, as the town in question still practiced segregation during the show.

Segregation of white people from those who were considered non-white was federally legal from 1896-1954 under a policy known as “Separate but Equal.” This denotes that as long as there are equal quality facilities or services for white and non-white persons, they are legally allowed to be racially segregated.

However, even though segregation was deemed to be unconstitutional in 1954 under Brown v. Board of Education, there were consistent attempts to maintain the policy in much of the United States, both through attempting to delay integration under local law, and through creating white specific spaces that were intended to excludes minorities. These programs existed throughout the United States and many were wildly successful up into the 60’s and 70’s.

So given those details, it’s important to understand the specific time and place that the orphanage is located in.

According to The Queen’s Gambit Wiki, Beth Harmon, the protagonist of The Queen’s Gambit, arrives at a fictional orphanage called the Meuthuen Home in July 1957. The home is located in Mount Sterling, Kentucky, which is an actual town located aprox. 15 miles (25 KM) outside of Lexington, Kentucky, with a 1960 population of over 60,000.

What is interesting is that the Fayette County School Board, in which Lexington but not Mount Stirling is located, began the process of school integration in Kentucky, admitting the first African American student in 1955. However, the Mt Sterling school system was one of the last to integrate in 1964 as a result of arson on the Dubois High School.

So, to answer your question, it is in fact historically accurate for an integrated orphanage to exist within both the United States and rural Kentucky in the mid-1950’s, but unlikely for the orphanage to exist specifically in Mount Stirling during the same period. The reality is that Methuen Home would have either existed somewhere else, been under a separate legal structure due to its religious nature preventing it from being held to the segregationist law, or have been within an area that the white community did not have regular access to, both socially and legally.

Edit: Grammer