The rules for Harpoon state that a critical hit activated the ability.

If a enemy ship which is harpooned has 2 HP left and is hit by 2 undefended hits and an undefended critical. Does the Harpoon ability still activate for the critical hit before it is destroyed (effectively hitting each nearby ship twice?)

It can only ever trigger once per application.

If the defender is critted, it triggers and is removed at that point.

If they are destroyed WITHOUT being critted, it triggers then.

It can never be activated both by crit AND by destruction.

A small distinction: The Harpooned condition specifically triggers after a ship is "hit" by an attack with an uncancelled critical result... NOT when the ship suffers damage or is dealt damage cards. This means it goes off before the damage from the actual attack goes through. Then, as part of that effect, the card is discarded, and the rest of the attack damage is dealt as normal (including the critical result). Even if the result of the damage destroys that ship, the Harpooned condition card has already been triggered, used, and discarded - no second wave of splash damage.

(Notably, if a ship has more than one Harpooned condition card on it for some reason, they are all triggered at once, by any relevant game state).

Indeed. Harpooned! will trigger off the uncancelled critical and it will then discard the card.

Which means that by the time you actually reach for your damage deck and realise you've been destroyed, you are no longer Harpooned! and it can't go off again.