Frankie Drake is the first that comes to mind. Agent Carter would also fit your criteria, although it's a very different genre.
Carnivale, Boardwalk Empire, the last season or two of Downton Abbey, Call the Midwife, Marvelous Mrs Maisel. All of those came up on search (amongst many more), and I enjoyed them. Mrs Maisel is one of my favorite shows ever.
I Love Lucy started in '51. Les Paul & Mary Ford at Home started in '54 (shorts, music & skits). The Honeymooners started in '56. The Donna Reed Show started in '58. Dobie Gillis started in '59.
Bear in mind that TV shows in that period usually displayed a highly idealized way of life (The Honeymooners less than the others). If you want to know what life was like in that period, Leave it to Beaver isn't going to help you. Books are your friend there.
- Boardwalk Empire - 1920s-30s period drama about mobsters, prohibition, etc set in New Jersey/Atlantic City
- Miss Fisher's Murder Mysteries - 1920's period detective show set in Melbourne
- Peaky Blinders - haven't watched it myself, but my understanding is it was pretty popular, 1920s period drama about a gang set in 1920s Birmingham
- Fargo (season 4) - haven't watched it again, but I've heard good things and season 4 (it's an anthology series, each season is a different cast/story so you don't have to have watched the previous seasons) is 1950s Missouri with gangs
The Untouchables has both options for you.
Movie being the most famous.
But there was also a tv series from 59-63 and a reboot 93-94.
Movie, not a show but Miller's Crossing is a masterpiece
lmao LA Confidential cast a shadow underneath Miller's Crossing
Babylon Berlin might be right up your street!
A Private Affair, Spanish PI show with Jean Reno as the trusty sidekick
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