We all know there are professions (fire fighters, police, soldiers, nurses, doctors, computer programmers, etc) where it is sometimes necessary to work extreme hours.One of those professions is not like the others.
And the question being asked is clearly, how do you keep operating when you have to work extreme hours?
And the honest answer is that, beyond a certain point, you simply can't. No-one is super-human. And clearly from the description the OP has given us, they have reached this point.
When you're being pressured to do those hours and you're already burnt out, it can be difficult to see your situation objectively and to make rational decisions about how to proceed. You are severely mentally impaired. The first step to fixing a problem is acknowledging that it exists, but in this sort of situation denial and misplaced faith that some sort of heroic effort will make everything better is all too common a response.
In reality, if taking enough time out to have a decent meal, a bit of exercise and a proper sleep before continuing will cause catastrophic failure of the project then the project was already in so much trouble that the OP should bear no responsibility for what happens next. Playing hero at the expense of their own health is most likely just delaying the inevitable.
Sure, there could be an exception here... But it's about as likely as winning the lottery every week for a year.