You might have guessed The Twilight Zone. But you'd be wrong. 

I thought it was an early 1950s show called "Tales of Tomorrow," which aired on ABC between 1951 and 1953. It was actually pretty good, but was performed live and you could often see stage hands roaming around on camera while the show was in progress. 

I was wrong about that one, too.

The answer is ""The Television Ghost," which aired over then experimental channel W2XAB (later WCBS) in New York City way back in 1931. Sadly, the footage is completely lost to history but the fact they were doing sci fi that early is simply astounding. It actually ran as a series, with a radio simulcast until 1933. The image below is the only thing that survives. 

What was it about? You can read about this obscure TV first here.