Can Baron Nashor be killed by Cull the Weak?

Yes, Baron Nashor can be killed by Cull the Weak.

According to the card text for Baron Nashor, it has the ability: "I can't be chosen by enemy spells and abilities."

However, Cull the Weak does not target or choose any units. As established in multiple RiftJudge FAQs (e.g., FAQ #8519, #8580), Cull the Weak instructs each player to kill one of their own units during the resolution of the spell. Because the spell does not "choose" a unit, it bypasses abilities that prevent a unit from being chosen.