Quitting undergraduate research internship
I think it would be a mistake to discuss your mental health. But you can more safely discuss the issues that might have led to it. It is natural to feel drained when family/relationship issues take so much mental and emotional effort. 

The best thing, assuming you cannot continue, is to discuss the underlying issues with someone involved in the program. They might have some solutions for you that you don't see now, such as deferring your participation. Be sure to stress your interest in the program and your abilities, so that you have some positive things to say beyond any obstacles in your way. 

If I were the person you talked to, I would try to find a way to make it possible. Your background suggests you would be the sort of person the PI would want to accommodate. 

The comment of user scrappedcola also seems wise.