The first thing you can do is engage in some serious educating of yourself. Humans aren't going anywhere soon unless something like an asteroid hits the Earth, but even that would be known well in advance. Yes, it's possible human life will disappear some day, but not for thousands of years. You won't be here whenever that happens. The Earth, on the other hand, will still be here and new forms of life will evolve. But as Mom says, you are going to die even if all humans don't die. We all live with this uncertainty. Intellectually, we face that by doing what we can to stay healthy or we try to die as soon as possible, which seems to be what people have done for most of recorded history by fighting wars. Truth is, life is the easiest and longest right now than it's ever been, and our ability to prevent catastrophe much greater. I would say, though, that this isn't something people haven't obsessed on -- all philosophy and religion is invented to try to deal with this thought. So you're not alone here. But worrying over it to the point you can't live the life you have isn't any fun.
I'm sorry you are having your brain intruded upon with these thoughts. That stinks. When we can't quiet our brain even when we know it needs to stop, it's challenging. How can you conquer this? Well, are you will to accept you have anxiety? You posted on the anxiety forum so that's a start. :))) But you should take it one step further. For you, it seems like talk therapy could be really excellent. Not to discuss the possibility of the world ending (hey, we could also get hit by a car tomorrow) but rather to discuss the anxiety. Why are you having this fear now? What can you do to cope with irrational thoughts and fear? How can you move anxiety to the side so it isn't interfering with your daily life. Maybe you'll end up needing medication to help but maybe not. But definitely, talking to a professional that can ask you key questions about the why, what and how of this will help. (again, not to pontificate the world ending but to discuss YOU and your obsession with it). Is seeing a counselor, psychologist, therapist, social worker, etc. a possibility for you or something you are open to?