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/.