It depends so much on context and various factors it's hard to answer this. What is "my best" though? If I'm not good enough to do something now, and its important enough to me, I can take steps to make me better. But not everything is worth that effort, and some things are risky for other reasons.
There is always value in doing the best one can, though, regardless of the results. But I'd go into different specifics as to how I'd personally feel about it depending on context.
As far as how I handle failure, though? I mean, that's heart breaking. As much as we all want to be healthy about it, it's also okay to feel shitty. I had a full blown freak out with two days of crying over two papers for school that both ended up getting As. (I did something wrong on one and it set off my anxiety). I've felt failure in the past too, and it's always awful to go through, but eventually you get back on your feet, you re-asses, and you figure out what direction is best. Maybe you can continue on that path, maybe it needs some adjustments, maybe a major change or shift is needed.
Specific to comics, I kind of designed my comic to be failure proof? My comic goals are 100% personal. My comic is designed to make me happy first and foremost. It actually was started during a period when I felt very bad about myself as a way of having something positive and structured to work on along with getting some other things in my life together. But I don't pursue it with a professional mind set specifically because of how painful and difficult that can be. I take it seriously, still, it's a legit passion, but I pursue it from a non-professional angle because I am better able to move forward that way.