I've just tested this out by holding a party in a de-pressurised weapons room. The crew seemed to have a great time and, providing the clone bay remained online and powered, all woke up the next morning with only minor hangovers.
By powering down the clone bay, however, as soon as a crewman died I was given a big angry warning saying "Clonebay off - Crew are dying!", despite the fact that I was able to re-assign power to fix this. When the last crewman's DNA was lost, I was given the standard Game Over screen
The clonebay will store a dead crew member's DNA whilst it has power. If it loses power for whatever reason, the warning will sound and you'll have around 4 seconds to restore power or lose that crewman. It does not matter how close they are to reviving, and once power is restored all progress is reset. This probably means it is possible to survive against non-drone non-lanius boarders for extended periods by de-pressurising the entire ship, letting everyone die, then juggling the power in your clone bay to keep your crew in storage, but stop them from reviving -- this idea is not without merit, since it's feasible that the invaders would die before kicking over the clonebay, and it would prevent your crew from constantly respawning and dying again, preserving their stats. Personally, however, if I were in this situation I'd just power down the clonebay and take the quick way out.
I suspect that the same will be true for enemy ships - you can capture and loot as soon as the last crewman's DNA is removed from the clonebay. How the Backup DNA Bank would change this is not clear.