What do you do with a captain card when it gets knocked out, or the hero is shipwrecked? Do they get taken out of the game or put in the nearest town's market stack?

I assume it goes into the nearest town's market stack. If you take dead captains out of the game eventually you would run out of captains and a player could end up stuck on an island with no captains.

So am I right? They go into the nearest town's market stack?

Ninjaman65 said:

According to the Rules, Captains that are discarded are discarded at the bottom of the Captain Deck. Captains are discarded in the following cases:

- a Hero hires a Captain while he still has another: he chooses which one to keep and which one to discard

- a Hero moves out of a Town with 2 Captains on the Town Market stack; he must discard one of his choice and leave only one Captain

- a Captain is killed; just like Allies are discarded when they're killed, they get to the bottom of their deck

You will never run out of Captains, as you will never run out of Challenges or Market Cards. With the exception of Encounter Cards that are removed from the game after resolving, all cards in Runebound are discarded at the bottom of the corresponding decks. Decks will never be exhausted, then.