Excellent performance by Nimoy as a possessed Spock, weak elements in the plot
While the performances, particularly that of Leonard Nimoy as the Henoch possessed Spock, are excellent, this episode suffers from the weakness of the plot. The Enterprise receives a telepathic message from a being called Sargon, whose intellectual essence is housed in a sphere 100 miles below the surface of the planet. Sargon, his wife Thalassa and his enemy Henoch are the last survivors of an advanced species that committed suicide via a fratricidal war. The mental essence of all three beings has been stored in the globes since the end of the war.
Sargon requests that he and his comrades be allowed to inhabit the bodies of Kirk, Spock and Dr. Ann Mulhall so that they can construct android bodies that will be their permanent residence. When all three agree to the transference the construction begins. However, Henoch has not changed, his goal is to kill Sargon and have Thalassa for himself. He also does not want to give up Spock's body and he tries to convince Thalassa to do the same.
However, Henoch is unaware of Sargon's true power, which allows Sargon to discover Henoch's intentions. In a ruse designed to convince Henoch, Thalassa and all on the Enterprise, Sargon makes everyone believe that Kirk's and Spock's mental essence perishes. In a last act of humanity, Kirk and Mulhall's bodies are once again inhabited by Sargon and Thalassa and they share a last kiss. Sargon and Thalassa leave their bodies in mid-kiss and the appearance of discomfort by Kirk and Mulhall after their departure is very well done.
The problem with the episode is that Sargon is portrayed as having so much power, certainly far greater than Henoch. With that much mental power, Sargon really would not have needed the bodies of humans in order to carry out his goal. Sargon would also have been wise enough to realize that with only three of the species left, the two males would be natural rivals for the only remaining female. The sensible solution would have been to provide another female for Henoch.

Pardon Me While I Borrow Your Body
This is the strangest and most atypical of classic Trek episodes. Neither Kirk nor Spock is the problem-solver. Their bodies are borrowed by advanced beings whose minds have been trapped inside crystal balls for half a million years. Understandably they are eager to get out of those balls and into bodies. In addition, guest-star Dianar Muldar lets her body be borrowed by a female alien, making three (Kirk, Spock, Muldar). The ending, in which two of the aliens depart, leaving Kirk and Muldar in a passionate embrace, is as "cute" an ending as you ever see. The joke is that this time, Kirk got the girl without even being inside his body!

Kirk's conscious contribution to the episode (that is, as Kirk) ends about half way through, when he gives the memorable "Risk is our business" speech. I tried to recall what it was like to hear this in the 1960s (I was a small child) and I think it was exciting... The space program was underway, everything seemed possible, and nothing else remotely like Trek was on the air.

The second half is fun, because the alien who gets to inhabit Spock is the bad guy, and Nimoy gets one of his rare opportunities to play a different (and very non-Spocky) personality. Spock's Satanic appearance is normally ironic, but here it's appropriate, because the alien inhabiting him is in this case so deliciously evil. It's evident that Nimoy, at least, is having fun.

There is possibly unintended Biblical resonance. The three aliens symbolically replay the Garden of Eden story, with Hennock (Spock) as Satan, Thalasia (Muldar's character) as Eve, and Sargon (Kirk) as both Adam and God. Sargon is also a Christ figure, for he "incarnates" as a human (Kirk) and later sacrifices himself for the good of the human race. (Remember that "kirk" is Old English for "church.") In effect, Shatner finally gets to play God. Make of it what you will.

Of note: Dianar Muldar also appeared in a third-season episode as a different character, but personally I find her far more fetching in this episodes where she wears the mini-skirt Federation outfit. (Yes, this is sexist, but the occasional non-PC quality of the original series is at times refreshing.) Also, the echoing effect used to enhance Sargon's voice whenever he is talking -- either as a disemodied voice (done by James Doohan) or as Kirk -- is extremely cheesy. But it was probably necessary, or else Sargon's disembodied voice would've been even stranger.

Not bad
In this offbeat episode, aliens change venue from giant spheres to crewmember bodies. Most of this episode lacks the action feel of many of its neighbors; it has an operatic, schmaltzy feel more akin to Metamorphosis or several 3rd season shows. Yet, I actually feel the episode takes a turn for the worse when the plot takes several predictably threatening turns. The action and drama are turned up, but at the expense of some of the episode's uniqueness; it becomes just another Enterprise in danger episode. It might not have been so bad to have the occasional feel-good show in which an interaction with aliens was synergistic from the get go.

Tidbit: This episode must beat out Spectre of the Gun for the 'longest teaser' award. Doohan, as was so often the case, played the voice of Sargon here. Muldaur would reappear in the superior Is There in Truth No Beauty?, as well as on The Next Generation.

"Risk is our business!!" ....that should be the catch phrase
I remember that line from this episode. Too bad the producers didn't use that catch phrase to promote the latest Trek movie. I can't remember such an unusual episode in a long time.

The best part of this episode is hearing James "Scotty" Doohan as the voice of Sargon. He's the evil alien entity who is left from a long dead civilization that makes us look primitive. It wasn't until I read the back of the cassette box that I discovered that Doohan voiced the ominous Sargon. Diana Muldaur; who portrayed Dr. Pulaski on Next Generation, plays Dr. Ann Mulhall. She was mysteriously assigned by Sargon who attempts to steal the bodies of Kirk, Spock and Mulhall. I know that Mulhall's name was not created by coincidence.

It has a little excitement. The scenes with a Sargon influenced Spock show an emotional side to Nimoy we hardly get to see. His alien counterpart is in love with the Mulhall. The takes with the artificial android bodies these other worldly beings were to inhabit were creepy by 1960s standards. The makeup artists had these flesh color duplicates of human bodies.... when activated would carry the mind of the aliens indefinitely. The sets were simple and very Outer Limits like in design. I wonder if Muldaur ever thought she'd be working on Star Trek again almost two decades later? This is one that can stand repeated viewing.

The Risk Soliloquy
Captain Kirk is at his most passionate and persuasive, as he answers Dr. McCoy's objection to a plan that involves
"tremendous" risk to the crew of the Enterprise. Kirk delivers a series of emotional yet convincing arguments, and finishes with a conclusion that proves irresistible: "That's why we're aboard her!"