Starfinder Core Rulebook mentions "archaic armor" in the "Archaic" weapon attribute description at page 180:
This weapon deals 5 fewer damage unless the target is wearing no armor or archaic armor. Archaic weapons are made of primitive materials such as wood or common steel.
This is the single mentioning of "archaic armor" (or maybe I haven't find any) in the rulebook. "Archaic" is a weapon attribute, not an armor attribute. The word "archaic" occurs once more in the context of armor, at page 196:
Modern armor is made of many different substances, including carbon fiber, ceramic, fabric, metal, and polymers. Most are constructed from a combination of materials, and some even use archaic materials such as animal hide.
If I understand correctly, this paragraph says that even a modern armor can include "archaic materials" like animal hide, but it is still considered modern armor.
So what is "archaic armor"? I can guess it could be any armor from the "old" Pathfinder setting, something like a chainmail, a cuirass, maybe even a XX-century bulletproof vest can be considered "archaic", but I have nothing in order to backup this assumption. A follow-up question — since archaic weapons are bad against modern armor, does archaic armor provide bad protection against modern weaponry? How bad?