Since the two Voyager probes launched in 1977 (Voyager I
and Voyager II
), it would be safe to assume that the fictitious Voyager VI launched in that time frame as well. 1977 + "more than 300 years" would suggest at least 2277+; So it's not too hard to extrapolate late 2270's or 2280s from that.
Except the movie came out in 1979 and there wasn't a third probe in development yet. Presumably at that time, the idea was that these were probes that would be launched in the future, not part of some alternate past. There's also the fact that the original "Grand Tour" plan that the Voyager program was scaled down from posited that the third and fourth probes would be launched in '79 -- and we all know how prone these things are to delays (V1 was originally proposed to be launched in '76 but went up in '77). If you look at the Mariner program
spun out of, the general pattern was two probes every 2-3 years.
So if we assume that in the Trek universe as posited by TMP, Voyagers 3 & 4 were launched in, say, 1980, that would mean a launch date of 1982-3 for 5 & 6.
Also -- that's only when it was launched
. Decker said it fell into a black hole more than 300 years earlier. There aren't any black holes in the known Solar system. V1&2 have been in flight for 35 years and they still haven't crossed the heliopause. Even if we assume some hitherto-unknown rogue black hole passed through the outskirts of the system, it still stands to reason that it was lost decades after it was launched -- sometime in the early 21st century at the earliest. Possibly much later.
So really, Decker's line would seem to place the movie squarely in the 24th century. It's not really reconcilable with our modern understanding of Trek chronology. The simplest explanation is that Decker misremembered his history lessons and got the dates wrong.
There would seem to have been more than a single "Eugenics War" in the Star Trek universe. In the Enterprise episode Hatchery, Jonathon Archer mention that his great-grandfather fought in the Eugenics War in Africa.
The episode was set in the year 2154, and Archer would have been about forty years old.
So a second Eugenics War around the 2080's sometime? Thirty years following the third world war.
It could work out, if previous generations had children late in life. There are 120 years between the start of the Eugenics Wars and Archer's birthdate. Add another 30 or so for Archer's ancestor's age at the time, and it can work if each of the three intervening generations had their children at around age 50. Not typical, but a lot of people have kids later in life these days. There are cases of people in their 60s or 70s having kids.
Then again, they were called the Eugenics Wars, plural, so that implies more than one conflict. Maybe the reason Spock said in "Space Seed" that they were the last of the world wars is that they kept being waged intermittently and eventually grew into WWIII in mid-century. Kinda like how some historians see WWII in Europe as a continuation of WWI.