Astronomy's still alive and well as a major course of study, but whether you can pursue a degree in astronomy will depend more on the college or university you're attending than whether the major exists at all. At places like University of Texas Astronomy still has its own department and degree plans.
To answer your question, though, it depends. Most astronomers aren't attached directly to an observatory. Most work at universities and use a variety of observatories around the world, depending on their affiliations, grants, etc. With some observatories this will mean traveling to the observatory to observe. With others the astronomers submit proposals to the observatory, and the observations themselves are made by a professional observer.
That being said there are still astronomer positions at most observatories. When applying for a job it helps if you already have an affiliation and a history with the observatory in question (e.g. having observed there or being a PI for a program there), but it's not strictly necessary.
As far as support jobs go (e.g. observers, optics, electronics, detectors, software, etc.) any number of degrees are useful. But there's still plenty of competition for these spots as well. An engineering degree isn't strictly necessary, but it helps to have one if you're working in an engineering field.
If you're a student something you can do that'll help your chances of getting a foot in the door is to do as many internships as you can find while you're in school. Don't be afraid to approach your department head or your advisor to ask for letters of introduction, information on existing internships, etc. Also don't be afraid to contact an observatory directly and express your interest. A number of our interns went this route, and wound up having some really productive internships.
Best of luck in your endeavors, stardust.