You can't cope as a human only being objective and rational. We have tried that for over 2000 years now and nobody have found the method to only live a life with objectivity and rationality.
Well, what do you mean by "objectivity" and "rationality?"
As far as I'm concerned, being objective means that you're focusing on facts rather than preferences, opinions, emotions, or judgments. It's objective when we say that an object is red even if we're wrong; it is subjective when we say that the same object looks terrible in red.
Objectivity is therefore not some holy grail of impossible standards. It doesn't necessarily mean that you're completely rational, unbiased, or omniscient. It just determines what kind of statements you examine and make.
As for rationality, most of the successful coping mechanisms I know of and have had success with are rational processes and include methods of becoming more objective. That's how we fight pathological ways of thinking like emotional reasoning and motivated reasoning, which tend to exacerbate mental illness and suffering in general. This has been successful for over 2,000 years, as early as Socrates at least, from what I know of history.
I'm not sure what you mean by "live a life with only objectivity and rationality." Do you mean live a life where you have no preferences or feelings whatsoever? Clearly, even desiring such a life would be a feeling, so that's impossible. However, in my opinion, it is completely possible to consistently live in accordance with "rational action" as defined by game theorists and decision theorists, which includes maintaining a perception of the world informed by objectivity rather than preference. That's literally living with only objectivity and rationality.
You could say that game theory and decision theory require goals and that these goals are therefore subjective. I would say that these goals are literally called objectives and you're being obtuse. You could say that the goals are not necessarily chosen through a rational process, but then I would say that there surely are goals which have been considered rational and that's the whole point of a number of philosophies, namely rationalism and intellectualism. Unless you're going to say that rationalism does not meet your definition of rational? But then, at that point, what can be called rational if we reject the definition given to us by the philosophy that has devoted the most time to rigorously defining rationality?
I would say that you can cope as a human only being objective and rational, and we found the method to only live a life with objectivity and rationality over 2,000 years ago. We have only refined our methods and understanding since then, but the core rational approach has remained the same.