Unlike the pirates on Barbossa's ship, The Black Pearl, the pirates aboard Davy Jones' The Flying Dutchman are cursed (or in better terms, bound in service, as it's not technically a curse like the Aztec Gold Curse), rather than undead. They can die, and we see many examples of them dying throughout the second and third movie - although they seem resistant to many attacks that would incapacitate or kill a regular person.
When a sailor dies at sea, Davy Jones will visit them and ask them:
Do you fear death? Do you fear that
dark abyss? All your deeds laid bare.
All your sins punished. I can offer
you... an escape-uh.
Life is cruel. Why should the afterlife
be any different? I offer you a choice.
Join my crew, and postpone the judgment.
One hundred years before the mast. Will
If the sailor agrees, Davy Jones will offer them a chance - to sail the seas for a fixed number of years (one hundred years) in return to their sworn allegiance to the Dutchman and its master.
A sailor must be dead or dying to receive this offer - they cannot be alive, as evidenced by Davy Jones' accusation of Will Turner:
You are neither dead nor dying. What
is your purpose here?
At the end of this service, they will pass onto 'the afterlife', and face judgement for their sins (perhaps at the hand of God) or more than likely - if they've gambled with their years of service aboard the ship - they will continue serving indefinitely.
It's worth noting that the concept of 'eternity' exists as well, as Bootstrap Bill Turner wagers an eternity of service in a game of dice against Davy Jones.
Bootstrap Bill, you're a liar and you
will spend an eternity on this ship!