One of the many notable devices in Star Trek is the transporter, which can beam people to any set coordinates. Has there ever been an instance in any Star Trek story where someone was beamed into something solid? For example, someone's arm being trapped in a table or something because they were accidentally beamed into it.
Per the Memory Alpha Transporter Section on Accidents there is no occasion listed in which there was an accidental beaming into another object.
That said, the closest such accident I would suggest would be the following:
In 2151, Crewman Ethan Novakovich was beamed back from the face of a planet later known as Archer IV by the still-experimental transporter system aboard Enterprise NX-01. The emergency transport was attempted during a fierce windstorm. Upon arrival, he was unconscious and had rocks, leaves, and other debris from the planet's surface embedded in his skin, due to a malfunction in the phase discriminator.
I would also suggest that the situation with the Pegasus embedded in Asteroid Gamma 601 was more of a cloaking device issue than a transporter. Hence, would not qualify as an answer to the question. But, would logically be a similar type of accident as would the Van Mayter accident noted above in the comments (which was due to a natural sub-space anomaly vs. technology issue).
Also see the Wiki page on Asteroid Gamma 601
In the TNG episode "The Schizoid Man" (season 2), an away team was beamed to a planet while the Enterprise was either still using warp power or had briefly dropped out of warp long enough to use the transporter. I forget the exact details but, after the away team arrives, Counselor Troi said something like, "For a moment I thought I was in that wall over there" and Worf replied, "For a moment, you were."
The possibility is mentioned:
In "The Cage"
ONE: Now, you all know the situation. We're hoping to transport down inside the Talosian community.
SPOCK: If our measurements and readings are an illusion also, one could find oneself materialised inside solid rock.
ONE: Nothing will be said if any volunteer wants to back out.
Repeated in "The Menagerie part 2"
NUMBER ONE: Now, you all know the situation. We're hoping to transport down inside the Talosian community.
SPOCK: If our measurements and readings are an illusion also, one could find oneself materialised inside solid rock.
NUMBER ONE: Nothing will be said if any volunteer wants to back out.
SARGON [OC]: I have locked your transporter device on my co-ordinates. Please come to us. Rescue us from oblivion.
SPOCK: Coming from deep under the planet's surface, Captain. Under at least one hundred miles of solid rock.
SARGON [OC]: I will make it possible for your transporter to beam you that deep beneath the surface. Have no fear.
SPOCK: Reading a chamber now. Oxygen-nitrogen atmosphere, suitable for human life support.
SCOTT: I don't like it, sir. The transporter co-ordinates preset by an alien of some sort. You could materialise inside solid rock.
MCCOY: Inside solid rock?
SPOCK: Unlikely. These co-ordinates correspond with the location of the subterranean chamber.
KIRK: I have a feeling that they or it could destroy us just standing here if they or it wanted to.
MULHALL: They or it?
And in "The Savage Curtain"
MCCOY: Yes, a big one. Suddenly, miraculously, we see a small spot of Earth-type environment down there. Now is it really there, or do we just think we see it down there?
SCOTT: You might beam down into a sea of molten lava.
KIRK: But why would he want to kill only two of us?
SPOCK: It would be illogical. With such abilities, they could as easily trick us into destroying the entire vessel.
So that is something that is definitely considered to be possible disaster.
I don't know why they don't worry about materializing in air and getting air bubbles in their bloodstreams.
And also in Star Trek II: The Wath of Khan
SAAVIK: This is not logical. These coordinates are deep inside Regula, ...a planetoid we know to be lifeless. KIRK: If Stage Two was completed, it was going to be underground. ...It was going to be underground, she said.
KIRK: Right! Let's go. ...Saavik. McCOY: Go? Where are we going? KIRK: Where they went. McCOY: Suppose they went nowhere. KIRK: Then this'll be your big chance to get away from it all.
Thanks to ClickRick for mentioning the last one.
While definitely non-canon, the "Choose Your Own Adventure" style book Star Trek: Phaser Fight includes a sequence that explores beaming into solid matter:
"Well, if you must," sighs Jinks. "The planet's surface is a bit tricky, so to be safe, you ought to beam down to the following coordinates: 33.51.62".
"That's solid rock," says Mr. Spock to the captain.
"I know," Kirk agrees.
You have the choice to go ahead and beam down at those coordinates.
"It's none of my business, sir", you pipe up, full of confidence, "but our readings were wrong when they told us that there wasn't any life on the planet. The readings are probably wrong about those coordinates being solid rock..."
But it doesn't go well...
You beam down to the surface of the planet. Or, more accurately, into the surface of the planet!
You beamed down into solid rock! The only part of you that’s visible above the rock is your hair!
Well, hair today, gone tomorrow!
It is a paradoxical question, if they have figured out how to transport someone without getting all their pieces mixed up with the atmosphere then they should not have to worry about mixing up their body with solid rock.
It should not be possible to transport into rock/stone because your body would have to have space to take up, the rock would have to move from the spot where you materialize, or be displaced. I think there would be some sort of feedback in the matter stream back to the source of transport if you tried to transport into something solid that could not be displaced by the introduction of another object.
Trying to beam into a location where the matter is not easily displaced would not be possible, The beam would be pushed back because the Rock/Stone would not give.
I would be more worried about beaming into Water or some sort of Gas or possibly even Lava as these substances will displace themselves around something being introduced into the midst of them, in other words these things will make room for the object being materialized.
The Matter Stream would not merge Matter from the stream with something outside of the stream. In the case of Neelix and Tuvok, they were both inside the matter stream(s) and materialized, from energy to matter, together, not into one another, from Energy to matter in the same space as Matter already existing in that space.
In the TNG episode "The Schizoid Man" (season 2), an away team was beamed to a planet while the Enterprise was either still using warp power, or had briefly dropped out of warp long enough to use the transporter. I forget the exact details, but after the away team arrives, Counselor Troi said something like "For a moment, I thought I was in that wall over there", and Worf replied "For a moment, you were."
This is because they were temporarily out of phase, which is different than the wall and the person existing, in the same place, at the same time, in the same phase, this is the same reason why the USS Pegasus had its incident in the asteroid, it was out of phase and came back into phase inside of the rock.
That is why they have the
Phase Discriminator. From its uses, it appears that it does two things
Makes sure that matter not being transported isn't merged with matter that is being transported
- In 2151, when Malcolm Reed attempted to beam Crewman Ethan Novakovich up from the surface of a planet during a powerful windstorm, Novakovich rematerialized with debris embedded in his skin, due to the transporter's phase discriminator's inability to isolate the debris from the matter stream. (ENT: "Strange New World")
Keep transported matter in the same "phase"(Time) from source to destination
So when transporting at near warp they existed through more than one "phase", so this is a little different than just straight transporting into Rock/Stone
With Pegasus, the ship and the asteroid existed in the same location but at a different time/phase and when the phase was normalized to the same phase as the rock there were no safeguards to keep the molecules from moving around and merging, which is why the accident happened.
I think that people in the future may be just as paranoid as we are, for instance if you were told to go into a cave without turning on your flashlight until you are already inside the cave at the predetermined location and then turn on your flashlight, you would probably not like it at all. so you would want to find out where you are beaming to and maybe see it before you just beam to that location. putting your faith in technology like this is going to be hard, I would assume that they would scan the location and probably not "beam" to a location they could not see. I also assume that their imaginations would run wild around the issue of beaming somewhere they could not first see, they would come up with all sorts of mythical issues that sound reasonable but maybe not possible, this would lead to the many discussions that other answers describe. This sort of thing happens today all over Social Media, there is no reason to believe that people of the future would not let their imaginations run wild, I know that most of these characters are very knowledgeable in the science behind the transporter, but I would say there is plenty that they don't know and plenty that they speculate on as well, causing the same imaginative ideas to pop up.
Not a personal accident, but the USS Pegasus was equipped with a phasing technology that failed halfway through an asteroid and made it materialize in solid rock.
Every time the transporter is used, it is beaming into an object. Air. Air is no more or less special than stone.
Since people don't come out of the transporter carbonated and bubbly inside the air must have gone somewhere. It is either converted into energy by the same process as the sending side, or some kind of force field protects the destination area.
I'm certain this occurred in an early episode of Enterprise where they used the transporter technology only for transporting cargo. They experimentally use it to transport people, and those people end up with bits of leaves and stuff sticking out of them but as far as I recall, no serious ill effects.
I vaguely recall that this has happened during transporter accidents, but I can't find an episode where it is actually shown, it just might have been somebody talking about it or something.
However, there definitely has been at least one case that we see onscreen case where multiple people are beamed into a solid object (a wall) although only temporarily. This is a side effect of a near warp transport, and does not appear to be in, any way, harmful.
In the non-canon Millennium trilogy of DS9 novels, Chief O'Brien is running a deep scan of the station when he turns up two dead Cardassians fused into a hull plate. After eliminating the possibility that they were sealed inside during manufacturing (according to Kira, even during the Occupation, Cardassian quality assurance would've picked up on something so obvious), he concludes that they must have been transported into the plate and that because of the sheer number of safeguards against such an accident, it must have been deliberate, i.e., murder.
Eventually, the crew determines that the incident was in fact an accident involving a time machine, which was in the process of powering down and so threw the Cardassians back a shorter time than expected. Instead of the room they expected, they appeared inside the wall due to the station's rotation.