That's kind of the dream for a lot of grad students, but I'm afraid the reality is fairly bleak. The vast majority of astrophysical research takes place in the context of tenure-track academic positions (or in the grad-student and postdoc steps leading up to that), which entails teaching; see my answer here for some thoughts on the difficulties even of that. Research positions without teaching requirements tend to be both less permanent (i.e., "soft money," i.e., you need to self-fund via successful grant proposals) and lower on the totem pole (i.e., under the direction of faculty--for example, research-associate positions). There are also staff members associated with observatories, but my understanding is that their roles are usually technical support (operations), not any kind of original research.
One of the few exceptions is (or was) Tony Tyson at AT&T Bell Labs, but he's the only one I ever heard of, and I rather doubt the position will exist after he retires. AT&T isn't really known for astronomy. ;-)