From a young age we associate quitting with being a loser. It's not okay to quit; it's not healthy to quit. Quitters don't go anywhere in life. Don't be a quitter.
Depending on the situation, that can be true. Sometimes, you need to give whatever situation you're dealing with a second (or third or tenth or hundredth) chance. Other times, it's okay to quit.
It's okay to quit an obligation that no longer brings you happiness. If you don't enjoy something, don't force yourself to continue and be miserable. Don't continue to make others happy. Ultimately, your happiness should be your priority.
It's okay to quit an obligation when others seem to continuously belittle your work and effort. I'm not saying it's okay to quit when someone tells you you're not doing your job; obviously, if you love your job and you work hard, continue at it. But if someone questions your work ethic and methods and overlooks everything you do, think about the situation. If your happiness does not require this obligation, quit.
It's okay to quit a relationship--whether it's a friendship, a romantic relationship, a work relationship, or any other relationship--when a person continuously brings you down. It's not healthy to surround yourself with negative people. Downers do nothing but bring you down, too. Sure, you will feel very odd for a while...you may wish you were still friends, best friends, or romantic partners, but eventually you'll realize that cutting ties was the best decision you ever made. It allowed you to be more positive, to enjoy happy times, to live a happier life.
I'm not writing this to tell you quitting is always the answer. Often times, it isn't. Sometimes you'll want to quit a responsibility or a job or a relationship because of the challenges you face, but usually those challenges lead you to growth and progress. It's usually not okay to quit to take the easy way out of things. It is okay, however, if you know you'll be happier, successful, and okay if you quit the specific situation you're facing.
It's okay to quit a job if you have another job lined up. It's okay to quit a relationship if you are not happy with the relationship, and you know you want to build your relationship with yourself or with others around you. It's okay to quit a situation even when those around you encourage you not to. Sometimes it's okay to be a quitter.
