Responsibility is one of the most loaded words around. "This is my responsibility, this is yours, you should take more, stop taking responsibility for that..."
So what is it truly about? As one of my teachers (Sadhguru) puts it so aptly: "Responsibility is the ability to respond." And it's up to you how you respond.
You see we cannot control our environment. Nor can we predict what will happen in any given situation. But what we can 'control' is how we deal with it.
The word control I always use a little more carefully because I know that some of us could use a little less control and more letting go haha. But by control I mean that once you've established a certain awareness of your thoughts, feelings and actions, you can decide how you'd like to act in a certain situation.
And that's why we can say: you are responsible for you own well-being. Regardless of what happens around you, it is in your ability to decide how you'd like to respond.
The responsibility is yours.