How do I cope with being hated for just doing my job?

You do your job right, and people resent you for it. That is a strange and lonely burden: the better you are at holding the line, the more some people dislike you for it. It is exhausting to be the one who says no correctly, day after day, and to absorb the anger that follows. But look closely at what the hatred is actually aimed at. It is not aimed at you. It is aimed at the limit you represent, the no they did not want to hear. If you were sloppy or unfair, that would be a real problem. Being disliked for being fair and consistent is not a failing. It is the price of integrity in a job that requires it. Still, you should not have to carry it alone. Do not let their reaction slowly convince you that you are the villain. Hold onto the people who know your character, and let the job be the job, not your whole identity. You can do right and be disliked in the same breath. Choosing right anyway is exactly what makes you worth trusting.