I think it's a little hard to say. If I had to guess this is sort of how and when I view the "barbarians" as existing:
The people who made the Barbarian set were from either before or after Rauru and Ganondorf's conflict at Hyrule's founding. I say this because the set itself bears no Zonai design influences (or at least not clear enough that I'd say they're contemporary) and look more primitive than sets like the Zonaite set. I think this can fit for both before and after because before Rauru's time (assuming we're still talking post-rest-of-the-zonai) the Hylians wouldn't have the benefits of Zonaite tech. After the Imprisoning War, this could also hold true though as much of the Zonai legacy gets lifted into the sky, covered over by the Hylians, and forgotten/twisted in history. Losing your God King immediately after they unify the land could put the land back into chaos and revert the state of the kingdom to how it was before where all the tribes are divided again and isolated. Without Zonaite and Zonaite tech as well as without the guidance of Rauru/Mineru, the Hylians are basically on their own again until they rise up and take control of the land.
On a completely different line of thinking though, there's no guarantee the Barbarians even were involved in the creation of the Zonai Ruins. Given that the Barbarian Set doesn't actually utilize Zonai designs or tech, this could easily just be a creation of primitive Hylians who settled in Faron post-IW, possibly after one of another Calamity we don't touch on in TotK (like the one before the Divine Beasts were created, if it exists).
If I had to commit to a view, I honestly want to say it's a mix of the last two. I think at the time of BotW there was definitely meant to be hints that the Lurelin folk were related to the people who made the Zonai stuff cuz of the shared crest. If we keep faithful to that intention, it would imply the Lurelin folk are related to the Hylians that built over the Zonai structures and built Thyphlo. However, given the actual lack of shared design styles between contemporary Zonai gear and the Barbarian set, I want to say that these people came far later than the era of the IW.
So basically you'd have descendants of the IW-Hylians (which in reality all BotW/TotK Hylians are descendants of) settling in Faron sometime after the IW and without the capability to create Zonaite gear in a time far removed from the age of Zonaite tech.
As I'm typing this though, I'm having sort of a change of heart about the Lurelin folk's relationship here. Given what I think about the relationship between the Ancient Sheikah and the Zonai (I think the Ancient Sheikah came almost immediately after Rauru's fall using information from Zelda's Purah Pad to know how to put all the Shrines and Towers everywhere as well as WHEN Calamities would appear) I don't think it actually leaves much room in history for the Hylians to revert back to a really primitive state. Zonaite tech was not LOST when the IW happened (yeah a bunch of stuff was raised but it's not implied they hid the tech or anything.), although the means they used to create it in the past may not be accessible anymore. Without Rauru and Mineru, they may have lost the ability to actually interface with the facilities used to create new Zonai tech. Zonaite itself would still be accessible in Hyrule though, meaning Hylians could begin their own independant studies into Zonaite and how to use it leading to Sheikah tech. This would all be happening in the same era that Hyrule Castle likely would've been built due to the relationship between the Zonai Ruins and the Sheikah Shrines, plus the Zonai Ruins being contemporary with Rauru's kingdom.
So let me try again: I think the people who wore the Barbarian Set were the Hylians of Sonia's era from before Rauru's intervention in bringing Zonaite tech and unity to Hyrule. After Rauru united the Hylians with the rest of the Hyruleans then gets sealed, the Hylians take the lead in ruling Hyrule which includes creating Hyrule Castle as a new capital directly over where Ganondorf is sealed. At this same time, a faction of Hylians who studied Zonaite and using the wisdom of the Zonai develop their own technology that we know as Ancient Sheikah tech.
