Near as I can tell it was "Happy Days" ( 1974-1984).
The show debuted in the 70s but depicted life (or at least a romanticized version of it) in the 1950s. Arguably the next "retro" show was "Laverne & Shirley" (1976-1983), another Garry Marshall production (which explains how Penny Marshall aka "Laverne" got the job) but as it was a spin off of Happy Days it doesn't really count. And there was "Joanie Loves Chachi" (1982-1983) which would have been a 60s show but it was also a spin off and seems to have wanted to remain set in the 50s despite the characters all being about 10 years older.
The next retro show, again near as I can tell, would appear to be "The Wonder Years" (1988-1993) which was set in the 1960s and seemed to grab a truer depiction of the times.
And currently we have "That '70s Show" (1998-Present) which seems to be more or less the Happy Days formula but set 20 years later.
So will there be an 80s show one day?