It goes off on the controllers turn the card says "After you refresh,"

And it does not matter if he is exhausted.

exhausted
A card that is exhausted (i.e. has one or more focus
tokens on it) cannot be focused to pay for card effects,
cannot focus to strike, and cannot generate resources.
However, other card abilities of an exhausted card
may still be triggered, and its constant effects (if any)
are still active, as long as such abilities do not require
the card to focus, to be ready, or are not otherwise
prohibited by the card text.

And it's a forced reaction, so it has to happen, even if it takes out one of your own units.

Relevant card text:

Forced Reaction: After you refresh, destroy the lowest cost non-Vehicle unit in play. (In case of a tie, the active player decides.)

This means, during your refresh phase, after you've removed 1 focus token from each card you control, you must destroy the lowest cost non-Vehicle unit in play. During your oppnent's refresh phase, you do not refresh your cards, so the Forced Reaction of Rancor will not trigger. However, even if Rancor is exhausted after you refresh, the Forced Reaction will trigger.

So if the Rancor is the lowest cost non-vehicle in play, will he eat himself?

I would think yes, but I wanted others' opinions.

I too would agree that if he was the lowest he would eat himself as it doesn't specify "Other" lowest cost non-vehicle.

Probably more like being the only one out there, desperatly seeking to devour someone and then unfortunately in despair tripping over his own feet, breaking his neck  

But yes, i would assume the same.

No assumption needed.  The rancor will indeed eat itself if there are no other potential meals of cost 4 or less available.

I maintain that starvation is the more thematic explanation, but now we're just talking semantics. Either way, Rancor is discarded when it's the last non-vehicle unit in play.

It will eat itself like Pizza the Hutt…

I wondered if someone was going to use the Pizza the Hut reference.  There it is.