In English, there's a phoneme commonly called "long A" (because it evolved from what used to be a lengthened /a:/). This part's pretty uncontroversial: it's the phoneme in the middle of "face".

However, linguists have different views on how to transcribe this sound. It's often pronounced as a diphthong, so some people write it as /eɪ/, /ei/, or /ej/; other people just write it as /e/ for simplicity, and say it's fundamentally a single unit. This mostly comes down to a transcription convention. One standard way to talk about it (in a purely English context), without committing to any particular transcription, is "FACE" or "the FACE vowel".

In the word "archaic", the underlying phonemes in question are this "long A" (the FACE vowel) followed by "short I" (the KIT vowel). Some people transcribe this as /eɪ.ɪ/ or the like. But for me, I certainly don't pronounce the same vowel twice in a row: the off-glide of FACE is higher than the vowel in KIT. So I prefer to transcribe it instead as /ejɪ/, to emphasize this height difference.

The same sequence appears in a lot of words, when a suffix starting with KIT is attached to a root ending in FACE: others have mentioned "mosaic" and its ilk, but it also shows up in "laying" and so on.

P.S. As for how it happened: note the diaeresis in the Greek ἀρχαϊκός, indicating two separate vowels /a.i/ with a syllable break between them. When it was eventually borrowed into English (via Latin and French), this was how it was pronounced; later sound changes turned /a/ into /ej/ and /i/ into /ɪ/.

I understand that essentially a word is "archaic" if it is old and not really used much today.

You'd be mistaken. The OED is correct in noting as the original and primary sense

Marked by the characteristics of an earlier period; old-fashioned, primitive, antiquated...

The important thing here is that it isn't about the quantifiable data on frequency over time.

The important thing is the way the word feels to most users. It's entirely arbitrary and subject to change.

Ethel might be centuries old, feel dated or archaic because it was more common among our grandmothers and then fell out of common use making us think of grandmothers when we hear it, and then return to fashionable current use when a pop star or anime trying to be archaic ends up making it cool and interesting again.

Archaic is steampunk for language. It's needless and kinda clunky, but it's not entirely obsolete because its clunkiness is endearing to at least some speakers.

More broadly,

esp. of language: Belonging to an earlier period, no longer in common use, though still retained either by individuals, or generally, for special purposes, poetical, liturgical, etc. Thus the pronunciation obleege is archaic in the first case; the pronoun thou in the second.

but again there's absolutely no objective criterion like you're asking for. It's only archaic when it feels that way and it stops being archaic once that feeling stops or once it drops entirely from regular use.

I wouldn't call "golly" archaic, but I would call it old and out of use

That's why most dictionaries have a dated note. You think ye olde England is entirely alien and its terms archaic but feel ye olde Americana is still relatable. That's entirely arbitrary. "Golly" is archaic to a zoomer, in the same way they have literally no sense of what a public rotary dial payphone is or why anyone would ever need one.

according to what authority

Well, the OED kinda earned that position, so them... but then it went digital and then it screwed up its user interface, so... not them or anyone else in the near future.