Real Bible Translation

Him who gave his own self beyond ourselves, so that He might ransom ourselves away from every lawlessness and might cleanse for his own self a people existing all around, zealous of good works!³

Julia Smith Literal 1876 Translation:
Who gave himself for us, that he might redeem us from all iniquity, and purify to himself a distinguished people, zealous of good works.

LITV Translation:
who gave Himself on our behalf, "that He might redeem us from all iniquity" "and purify" "a people" "for His own possession," zealous of good works. Psa. 130:8; Eze. 37:23; Deut. 14:2

Footnotes

Existence surpassing the normal bounds

If we don't go by idiomatic usage (since we are after the real and not the idiom or metaphor), the literal and morphological sense is:

περιούσιον = περί (“around, beyond, over”) + οὐσία (“being, essence, substance, property”).

περί in compounds can mean around spatially, but very often it carries the force of beyond, exceeding, in addition to.
οὐσία from the verb εἰμί (“to be”), originally “being, existence,” later “substance, property, wealth.”
Morphologically, περιούσιος would mean “being beyond” or “over-being,” i.e., “that which exists beyond the ordinary” or “surplus/exceeding essence.”

So, if you take it in the most literal, non-idiomatic sense, περιούσιον could be described as:

“beyond-being,” “super-being,” or “that which exists over and above.”

From there, the idiomatic “special possession” is just the conventionalized meaning — but the root picture is of existence surpassing the normal bounds.