Duty is simply an inherent responsibility we have to the world in which we live to fulfil our purpose for being on the Earth. Stoics believe that humans are meant to be rational and social beings; therefore our primary purpose for existing is to be the best rational person we can and to benefit society; that is, to live in accordance with nature.
There are aspects of stoicism that may appear hypocritical or contradictory; such as a technique for avoiding fear and anxiety would be to avoid circumstances that cause that. That may lead one to become a hermit and swear of human interaction. It is our duty; however, that keeps us from doing so. We are meant to be social animals, so when you can’t avoid the circumstances that cause negative emotion you should evoke the strategies for coping with them—fatalism, duty, cosmic scale, etc.
The stoics, as revealed in fatalism, believed in the fates, as well as Ancient Greek and, later, Roman gods. They had a convenient source to draw strength from; the gods made them to be rational and stoic, so they better live up to that standard.
I say, regardless of your religious orientation, you should be able to rationalise that it is in our nature to be both rationale and social, and that these characteristics are deep and primal to our being and we should foster them. That, at least, is how I justify my purpose with my agnostic belief system. More on that later.