I think that was the show Dark Matter.
There is an episode of ST:TOS with long-distance transportation - Gary Seven’s people can transport people hundreds of light-years.
So can the Metrons in “Arena.”
Yep, “Contagion” is the episode I was thinking of. I wonder if it was the first to use the trope “Have you tried turning it off and then back on again?”* 
*I call this the “Howard Wolowitz maneuver.” 
I had forgotten that - I remembered the Gary Seven bit, because it’s explicitly recognized by the Enterprise crew as a form of transporter, like theirs, but way better.
Yep. The Metrons just blinked and it was done. No machinery involved, SFAWK.
I think the Organians had the same capability in “Errand of Mercy.”
That story is remarkably prescient for 1877, since Einstein didn’t mathematically prove the existence of atoms and molecules until 1905. Maybe some real time travel was at work here? 
https://www.ans.org/news/article-969/albert-einstein-and-the-most-elemental-atomic-theory/
I also wonder if this wasn’t the inspiration for the TNG two-parter where Data’s 500-year-old head is discovered by an engineering team in San Francisco.
Atoms and molecules had been discussed and theorized about since the 18th century. They were a working theory for scientists, especially chemists, for a long time.
Boltzmann was the main proponent of atomic theory at the end of the 19th century, but some scientists were still sceptical until Einstein’s 1905 paper.
Couldn’t we go back even farther than that?
Right. For that matter, atomic theory went all the way back to Ancient Greece, but it was still hotly debated until 1905. I watched a bio of Einstein not too long ago in which one of his professors castigated him for supporting the molecular theory.
That’s ‘philosophical’ atomism. As a scientific theory to explain chemical reactions and thermodynamics, it was developed in the 18th century. See the wiki on Atomic theory.
Pun intended?
Sort of. 
Like the jinn depicted in One Thousand and One Nights 1000+ years ago.
SFDebris recently reposted his review of this, and he comes up with a reason. The previously aired episode was “The Naked Time”—you know, the one where everyone got drunk with “heavy water” and were just generally messing everything up. So he says (in a facetious tone) that they must’ve sabotaged the shuttles and they weren’t repaired yet.
He also notes that, in reality, they just hadn’t invented the shuttlecrafts yet.
There’s Rogue Moon. That’s pretty serious, and has the identity issue also.
Even more worrisome:
This was on a Trustees’ day, and I was the only visitor in the hall. The faithful attendant
had gone to enjoy a can of beer with his friend, the superintendent of the monkeys.
Beer cans weren’t invented until 1933!
Could simply have been a canister of beer that he brought to work with his lunch, maybe one like this:
Which is interesting, since the Enterprise was designed to have a shuttle bay (in the rear of the secondary hull) right from the start.
In this case “hadn’t invented yet” means “hadn’t had good enough reason to build the prop yet.”
The engineering set wasn’t built until “Enemy Within” - and I guess the shuttle sets were in line after that.