It is a common assertion that the "United Starship Republic, number 1371" was a Constitution Class ship like the Enterprise, NCC-1701, based on the term "Starship". We might also note that the name "Republic" fits nicely in the naming scheme of a class named "Constitution".
Beyond that, however, nothing points to the idea of the Republic as a Constitution Class as having any merit, and I thus propose that this is not so.
We first hear of the ship in "Court Martial" simply as a vessel to which Kirk and his old friend, former Academy instructor Finney, had been assigned "some years" after their time at the Academy together. This suggests the ship was active, likely in the late-2250s.
The only other detail we hear about is that the ship had "atomic matter piles" aboard. Per Memory Alpha, this terminology should be associated with "atomic piles", an early term for a fission reactor based on the stacking of uranium and graphite moderator in layers. It is also possible that this is related to the "energy pile" aboard the Antares from "Charlie X"[TOS1], though these two thoughts are not necessarily contrary to each other.
Star Trek IV would seem to counter the hypothesis from Memory Alpha. Spock notes:
"If memory serves, there was a dubious flirtation with nuclear fission reactors resulting in toxic side effects. By the beginning of the fusion era, these reactors had been replaced, but at this time, we may be able to find some."
This would seem to make "atomic matter piles" as a fission based power source (even on an auxiliary level) dubious at best.
confused with Finney's apparent Atomic Doobie Pile.
Still, it remains the case that the Antares was an old ship, and, even if fission is unrelated to the "atomic matter pile", it seems to also be an old system of a sort never heard of again, and notably not heard of aboard the Constitution Class Enterprise.
The Republic was still active in 2293, her name, registry, and Neutral Zone patrol mission visible on a chart titled "Starship Mission Assignments". At this point the ship was at least about 35 years old and likely very much older, given her low registry.
Two other details now become relevant.
First, we have Picard's note to Scotty in "Relics" confirming familiarity with the Constitution Class based on the fact "there's one in the fleet museum". Based on the conversation, this was clearly not an Enterprise, regardless of any later alternate universe confusion. If the Republic was a Constitution Class ship then it plausibly could've been the one in the museum, but that brings us to the next point.
That is, second, we hear of a training ship Republic in a conversation aboard the Valiant between a Red Squad cadet Collins and Jake Sisko. She notes that they were on a training cruise, leading Jake to ask:
"So this was a training ship? Like the other one, the, er, the Republic?"
The cadet replies:
"Not quite. The Republic's an old ship. I don't think she's left the Terran system in fifty years."
This puts a ship named Republic as an old ship that hasn't even left the solar system since circa 2321.
It is, of course, entirely possible that the Republic 1371 was retired shortly after 2293, or even destroyed on the very Neutral Zone patrol we hear of, with the name quickly recycled onto a new ship. This new ship, for whatever reason, doesn't manage to have nearly so long a service life, relegated instead to a training ship sometime prior to 2321, by which time its advanced age seemingly precludes travel beyond Earth's solar system.
However, that doesn't seem to make a lot of sense, prima facie.
(Oh sure, we can work up a story where another ship gets relegated to training, easily enough . . . maybe NCC-1371 was retired and the new Republic, NCC-3171 or whatever, was built a few years later, perhaps even of antique parts. Insert your own concept here . . . maybe it was of an unknown and quickly retired class (but one similar enough to existing ships to warrant keeping it), or maybe after the Tomed Incident in 2311 or just some other random episode of bad luck on a science mission she was no longer capable of high warp due to spaceframe damage. Rather than scrap a relatively new bird, she became a training ship, and off we go. It's plausible enough, albeit rather ad hoc.)
Despite some creative possibilities, it seems much more likely that the training ship in question is the Republic, 1371. Given Picard's commentary, and given that his Academy years would've been circa 2323 to 2327 (around the last time the Republic left the Terran system), Picard would likely have trained on the Republic. If she were a Constitution Class ship, then it would've been relevant to note the training aboard one to Scotty rather than make him feel like a museum piece, himself.
(One could here argue that a training ship and a museum ship need not be mutually exclusive concepts, and thus that the training ship Republic and the museum Constitution are the same ship. While technically true -- a few US Navy sailors train on the sailing vessel USS Constitution, for example (which is still in active commission, unlike the typical museum ship) -- that's a weak argument. A ship identified as a training ship and a ship identified as a museum ship are functionally different concepts, even if training occurs on some museum ships. Like the Republic replacement hypothesis, this is ad hoc.)
So, all indications are that the Republic 1371 still exists as of 2371 as an Academy training vessel. And, thanks to Picard's commentary, it is apparent that she's not a Constitution Class vessel, but another ship type entirely.
Alas, we have no way of knowing what kind of vessel it was. The fleet between the Federation's founding in 2161 and the launch of the Enterprise in 2245 is largely unknown. By certain averages and the registry number, I would propose a construction date for the ship of circa 2223, but one's mileage may vary. Thanks to the Constellation 1017 and possibly Eagle 956, we know that the ship was a rough contemporary of some early-build Constitutions, but that's about all we know.
We'll explore the possibilities in another post.