If I play Baron to a battlefield and my opponent has an Apathetic Vex, he gets stunned. Does he go back to base?

No, Baron Nashor does not go back to your base.

Reasoning

1. Stun Effect: As established by RiftJudge FAQ #9240 and FAQ #8685, Vex, Apathetic's ability is a triggered ability that programmatically selects the unit you just played. Because it is programmatic selection rather than "choosing" or "targeting," it bypasses Baron Nashor's protection ("I can't be chosen by enemy spells and abilities").
2. Stun Mechanics: Being stunned renders a unit "Stunned" (Rule 423.1). Stunned units are restricted from contributing might to damage in combat (Rule 423.1.b) and gain a restriction against movement for the turn (per the text on Vex, Apathetic).
3. No Recall: Stunned units remain on the battlefield. There is no game rule or mechanic that causes a unit to automatically return to its base simply for becoming stunned. While combat rules state that attacking units are recalled to base during combat cleanup (Phase 2, step 2 of <showdown_flow>), simply playing a unit (even if it enters an occupied battlefield) does not inherently initiate combat that would result in a recall to base unless a showdown/combat is specifically triggered. Even if combat were triggered, the unit would remain at the battlefield until the combat cleanup step, at which point it would return to base as part of normal combat resolution—not because of the stun itself.

Summary: Your Baron Nashor will remain on the battlefield, stunned and unable to move for the remainder of the turn.