When did monsters get trapped underground in Undertale?

There are many clues but no one has confirmed when the barrier was created

TL;DR: Circa 1990-2009, assuming that Gerson is not much more than ~120 years old. But we don't actually have evidence of Gerson's age, so it could be longer than that. Possibly a lot longer.

I am going to disagree with the accepted answer here. There is actually quite a bit of additional useful information in Undertale which, taken together, paints a vague but not completely opaque picture of the overall timeline here. Let's go through some specifics:

• This timeline posits that it was "sometime before 100 AD" on the basis of a single line from Bratty in the post-True Pacifist epilogue (referring to "millenia of being imprisoned"). Personally, I'm inclined to view this conclusion with skepticism, because Bratty and Catty are prone to speaking hyperbolically and I'm not willing to take this line completely literally (i.e. "millenia" = "at least 2000 years"). But we can raid that timeline for other evidence, much of which is more robust than that.
  • In fairness to the person who created that timeline, the note underneath explicitly acknowledges that "millenia" might be hyperbole.
• Toriel and Asgore are both unaging (according to Gerson's post-True Pacifist dialog). So we can't use their apparent ages to conclude anything. However, this dialog also implies that this property is exclusive to Asgore and Toriel; other monsters age normally.
• The monsters initially lived in the city of Home (which is now the Ruins). It is strongly implied that Toriel, Asgore, and Asriel all lived in Home together, in the building which is now Toriel's house. For the bulk of this period, Chara was not yet present, so it was mostly or entirely pre-201X.
  • Sources: The history book on Toriel's bookshelf tells you about the monsters living in Home. Toriel's house has three chairs at the dining table, one smaller than the other two. Additionally, the house has a child's bedroom, with one bed, as well as some striped shirts and other items which may have belonged to Asriel. There is a conspicuous lack of evidence of a second child living there.
• At some point, the monsters decided that they "fear[ed] the humans no longer," and moved out into the rest of the Underground, ultimately settling in New Home. The royal family, now consisting of four members, lived in New Home. So this was post-201X (Chara had been adopted by the Dreemurrs), and Chara's presence might be the reason that monsters stopped fearing humans. Regardless, Chara was widely known to the monsters of the Underground, and (in general) the monsters approved of the royal family adopting them.
  • Sources: The history book in the Snowdin library mentions "fearing the humans no longer" and resettling in New Home. It also mentions Snowdin, Waterfall, and Hotland in passing (but without giving their names). The house in New Home has four chairs at the dining table, of which two are smaller. It also has two children's beds, in the same bedroom, and a number of items which probably belonged to Asriel and Chara. Finally, the story told in New Home emphasizes that Chara's presence caused the Underground to be "filled with hope."
• The previous two bullets tell us that the war was probably not much later than the 2000's or so (Asriel was old enough to sleep in a regular bed, sit in a regular chair, etc. when the royal family left Home - and he was also old enough to help Chara walk in 201X, as seen in the True Pacifist ending cutscene). So now the question is, how early could the war have been?
• The overall amount of time that elapsed between the war and the events of Undertale cannot be too lengthy, because Undyne claims (over the phone) that Gerson fought in said war. He's still alive (if rather old).
  • Several monsters have UnderNet usernames which are suggestive of their birth years. In particular, Papyrus is CoolSkeleton95 and Undyne is StrongFish91. If those are birth years (which seems likely to me, but might be debatable), then we must either assume that Undyne and Papyrus were born in 199X (and were around for all seven humans that fell before the events of Undertale), or that they were born in 209X (so Chara died long before they were born). Undyne's dialog on meeting the protagonist in a neutral route suggests that she has never met a human before, favoring the latter interpretation. This also tracks well with most of the monsters in the Underground failing to explicitly identify Frisk as a human (e.g. the shopkeeper in Snowdin), with Sans's (possibly hyperbolic!) claim that Toriel hadn't heard a good joke in "a hundred years," and with Toriel's claim that Frisk is "the first human to come here in a long time." All of those things together point to a very long timeline, which the 199X interpretation doesn't give us.
  • Many decades have elapsed since 201X. Either Gerson is positively ancient, or Toriel and Asgore had Asriel shortly after the war. Allowing for time for Asriel to (partially) grow up, and for Undyne and Papyrus to (fully) grow up (and assuming that all three monsters age at the same rate as humans), we've already accounted for well over 100 years of Gerson's life. So I'm disinclined to put the war too much earlier than 2000, maybe a decade or so at most.
  • However, this is a matter of interpretation as we don't strictly know how long Gerson can live (or indeed, how fast monsters age in general). In Deltarune, neither his tombstone nor his memorial bench have a birth or death year on them (but their presence suggests that he is at least capable of dying of old age, since Hometown is not exactly a violent place).