Which one is correct?
1. I cannot do not eat for 5 days straight
2. I cannot not eat for 5 days straight
3. I cannot not to eat for 5 days straight
While it is possible to say "I cannot not eat..." You should avoid this construction if possible. In this case you could say "I cannot fast for 5 days" or "I cannot go without food for five days".
Other words don't have convenient ways of expressing the negative "not eat" with positive verb "fast". However it is usually possible to say "I cannot stop breathing for five minutes", or perhaps "I cannot give up jogging for five months."
There is no expression "I cannot not X" that can't be better expressed in a way that avoids "cannot not".