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.
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?