Came from a poor background, had an unhappy childhood, was bullied, was abandoned by his only friend (who went off and married one of the bullies), accidentally betrayed her and was responsible for her death, spent the rest of his life trying to atone, doing a job he hated, in the place that reminded him, every day, of his unhappy teen years and the woman he had lost, was forced to grovel to two masters, was royally pissed-on by both, had to save the life of the boy who looked just like the bully, was forced to be polite to the werewolf who almost killed him, was made head of the most despised house in the school and was expected to look after a load of misfits, was violently murdered by the fucking symbol of his house, and generally raised being thoroughly miserable to the status of art-form.
Yup, I think Severus Snape does angst. And is entitled to whine a bit. Or a lot. In very long and erudite sentences.