Background music, Perry Como's "Catch a Falling A Star" was released in 1957. This series takes place in post-WWII 1940's.
In the establishing shot outside the Western Costume Co. there is nothing written on the side of the marquee to the left of the building main entrance, however after a car drives past (across the frame from right to left) the writing CHATEAU BAL has appeared on the side of the marquee, within the same shot.
Aspiring screenwriter Archie Coleman refers to himself as "black". However, in the era the series is set in the term "black" would not have been used, as African Americans would have been referred to as "colored".
When Jack lights Archie's cigarette in the coffee shop, he uses a disposable lighter that would not be invented for another 30 years.
Ostensibly this takes place shortly after the end of WW II. Perry Como's "Catch A Falling Star" was not released until 1957.
When Jack and Henrietta are in the gynecologist's office, the sign behind Jack that says "Congenital Syphilis is Preventable" is labeled "New York State Department of Health," but the show takes place in Hollywood, California.