Your last question first: no, save in this sense is not archaic. It’s not as common as except, but it occurs in natural speech, especially as part of the phrase save for.

As to why save (for) means ‘except (for)’, that is a relatively long story.

Originally, in mediaeval French, sauf/salf (masculine) and sauve/salve (feminine) were used as adjectives meaning ‘whole, intact, safe’ only. All the way back since Latin, this adjective was frequently used in an absolute manner to mean ‘with [X] safe/whole/intact’: in Latin, this was an absolute ablative, and you had expressions like salvā fīdē ‘with your faith intact/without violating your faith’. In French, where the ablative was long gone, the absolute usage was simply just the plain form, but still originally inflected to fit the number and gender of the noun it qualified: sauf sun ordre ‘with his order intact/not disobeying his order’ vs. sauve sa grace ‘with his grace or lordship intact/not disobeying or disrespecting (etc.) his grace or lordship’.

During Middle French, the inflection to match the noun modified started to be lost, and the masculine singular form sauf was used, becoming more and more preposition- or conjunction-like.

The meaning also shifted: from indicating that something was kept intact and unviolated, it came to be used as a polite way of expressing disagreement, similar to pace in modern academia: it meant something like ‘while not meaning any violation towards X’, ‘if X will excuse me saying so’, etc.

From this meaning came a more abstract meaning of the object of save, which was now clearly a preposition. This object now started being considered as having been removed from harm’s way, having been taken apart to avoid violation or unwholesomeness, as it were. The thing that was ‘safe/save’ in this sense was therefore now simply considered as not being included in the main group. The overlapping sense can be envisioned in an example phrase like “All the men were killed in the war, save one”: the one man is removed from his group of belonging (“all the men”) and is considered apart, immune to the effects of the sentence upon that group. He is both literally safe from being killed in the war and ‘safe’ from whatever happens in the sentence, so to speak.

This nuance of meaning was probably blurred out quite quickly, and only the notion of the object being seen as something that had been taken out of its group and was seen as an exception survived, which is basically the meaning both save and except has today.

[This answer is based on and fleshed out from the definitions and etymological notes given in the OED article for save (requires subscription).]