does cull the weak kill baron

Yes — Cull the Weak kills Baron Nashor.

Baron's "I can't be chosen by enemy spells and abilities" only blocks effects that choose/target it. Cull the Weak doesn't choose anything: it just instructs each player to kill one of their own units during resolution, so the protection never applies (FAQ #9385).

One nuance: since the kill is the Baron controller's own choice, if they have other units they could pick one of those instead — but Baron is a legal choice regardless of its ability.