I assumed that the
Enterprise had already crossed the galactic barrier safely.
I assumed that whatever Kirk called the farthest star was behind them or at least closer to the main body of the galaxy than they were.
But of course it couldn't really be exactly the "farthest star" in the Milky Way Galaxy, not if that meant farthest from the galactic center. .There are certainly extremely thinly scattered stars and star clusters out to distances of perhaps a hundred thousand light years from the galactic disc. Questar M-17 itself should be a dead star, as Henoch says, so it should be an example of a star "beyond the farthest star" and thus even farther from the main body of the galaxy than the allegedly "farthest star".
Or maybe the
Enterprise had crossed the galactic barrier and reached the outermost and farthest star and then turned back toward the galaxy to visit Questar M-17, which is closer to the main part of the Galaxy than the farthest star is, but is farther on in the voyage of the
Enterprise and thus counts as being "Beyond the Farthest Star" according to the path of the
Enterprise
As for star charting when the
Enterprise is "Beyond the Farthest Star", people seem to assume that a starship can only chart stars that are ahead of it. But a starship can chart stars that are ahead and behind, right and left, above and below, in all directions. So possibly the
Enterprise could be charting stars that are back in the main part of the galaxy from a different direction than they have been charted before.
And some people may assume that a starship can only chart stars that are near it, within a specific limited distance. Actually it is possible to chart the positions of stars which are thousands and millions of light years away. Astronomical instruments on Earth or in Earth's solar system have detected individual stars thousands and millions of light years from Earth. The farthest individual star detected,
https://en.wikipedia.org/wiki/MACS_J1149_Lensed_Star_1 was about 9.34 billion light years away when it emitted the light that was detected, and is now 14.4 billion light years away due to expansion of the universe.
There are two forms of star charting. One is recording the apparent directions to the various stars as seen from a specific location. Astronomers in the 19th, 20th, and 21st centuries - let alone astronomers in the 23rd and 24th centuries, could already do that with incredible and ever increasing precision.
The second form of star charting is determining the actual positions of stars in three dimensional space, and for that both the direction to the star and the distance to the star are needed.
So how can the distances to stars be measured? The direct way to measure the distances to stars is measure the difference in the directions to them from two or more different positions. For centuries astronomers have been measuring the directions to stars from positions two Astronomical Units apart. That is about 298 million kilometers 185 million miles apart. How could astronomers make measurements that far apart before the Space Age? By simply making their measurements six months apart, from positions two Astronomical Units apart on Earth's orbit around the Sun.
By so doing astronomers have measured the distances to thousands and millions of nearby stars. And that is using a baseline only 2 Astronomical units long. Taking measurements 2 thousand or 2 million Astronomical Units apart would make the measurements a thousand times or a million times more accurate.
Thee are 63.241.077 Astronomical Units in a light year. Alpha Centuari, the closest star to the Sun, is about 4.3 light years from Earth. Rigel is about 880 light years from Earth and Deneb is over 2,000 light years from Earth.
So when the
Enterprise went "Where No Man Has Gone Before" in that episode and crossed the galactic Barrier it could have charted stars from a position hundreds or thousands of light years farther from Earth than had ever been done before, and so when its data would be combined with data from other places it would make the position calculations for thousands or millions of stars much more precise than they had ever been before.
And by charting the directions to many stars in the galaxy for a new angle when the
Enterprise left the galaxy again and went "Beyond the Farthest Star" the new data would help to make the three dimensional positions of many stars even more precise.
; 
;
