Odds of Getting In
It's all about getting into a school for a bachelor's degree. And then another school for your master's. And then another for your Ph.D. And then you may need a postdoctoral...or two...or three...before getting hired in a permanent position.
The competition for permanent research positions, like those for universities, is stronger than Mike Tyson (no relation to Neil Degrasse), and job growth for astronomers is rising slightly below the average rate for other jobs. Not too promising.
For other research jobs, you'll be relying on federal funds and grants in order to do your thing. The competition for this is still strong, but only as strong as a division II school's quarterback. You'll constantly be searching for another job every couple of years when each project ends. And however much money the government has is going to affect how much money you have. There are a lot of things that can get in the way of an astronomer and his or her reach for the stars (source).