Starfinder Core Rulebook mentions "archaic armor" in the "Archaic" weapon attribute description at page 180:
Archaic
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:
Armor
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?
Best Answer
Forward compatibility
Paizo developers have recently (I believe it started with the Advanced Class Guide) to use a development strategy that they have called Future Proof (known in programming as Forward Compatibility, as opposed to Backwards Compatibility), which means they write rules that will make future products compatible with past products.
The first examples of that being deliberately done (that I am aware of) was the Improved Spell Sharing and Pact Flanking feats from the Advanced Class Guide. Both feats, as written, would not work and would not do what is written because nobody else other than a Hunter could qualify for their requirements. For teamwork feats to work, both creatures must qualify for their requirements, and animal companions cannot own animal companions. So it only works for a class like the Hunter, which can share teamwork feats with her companion. But the wording of Improved Spell Sharing allows, for example, that a future summoner archetype share teamwork feats with her eidolon and still benefit from this feat.
But there are instances of such a strategy being used back in the Core Rulebook, when we see texts like "or similar effects" (See Haste). By the publication of that rule, they do not know all possible similar effects, so they use a catch-all term that could be referred back to in future products.
Another good practice we can notice in recent books (like Occult Adventures) is to not tie a feat to a certain class (as can be seen in the core), which later would have to be errata'ed. Instead, the requirement is tied to a specifically named class feature, and as such, they can simply say "works as that class feature" and all past products are automatically compatible. With this, they avoid pitfalls like what happened with Improved Familiar.
For Starfinder, they have realized that there will be rules further expanded in future books, and left rules in the book that will not be of much use yet, or simple rules for something that we know that should have been more complex, like listing only three different drugs in the core rulebook.
You should expect to see Archaic armor properly defined in future NPC sheets or supplements, and I suspect you are correct, they must be talking about Pathfinder style armors and weapons.