If the Earth ship Valiant's impulse engines were anything as fast as the Romulan impulse engines in "Balance of Terror" scaled against that map with a 1 LY wide Neutral Zone, it's very possible for the Valiant make it most of the way to the edge of the galaxy in a fairly short amount of time. Although based on the recorder buoy, it didn't sound like they intended to go out of the galaxy.
Without knowing what Kirk felt was "impossible" it is open to interpretation as to the Valiant's capabilities.
Or was it "impossible" to think that another Earth ship already probed out of the galaxy before the Enterprise did?
Without knowing what Kirk felt was "impossible" it is open to interpretation as to the Valiant's capabilities.
Captain's log, Star date 1312.4. The impossible has happened. From directly ahead, we're picking up a recorded distress signal, the call letters of a vessel which has been missing for over two centuries. Did another Earth ship once probe out of the galaxy as we intend to do? What happened to it out there? Is this some warning they've left behind?
Was it "impossible" because the ship had been missing for over two centuries?
Or was it "impossible" to think that another Earth ship already probed out of the galaxy before the Enterprise did?

That is to say, there are no cruisers or tugs or science vessels or hospital ships. This is all Earth of the late 21st century could muster; a general-purpose modular FTL spacecraft that could swap out ring pods for mission-specific applications. (Not unlike the Eagles of SPACE: 1999)