Is astrophysics a good career field to try to enter in the 21st century?

Your concern is entirely valid; astrophysics is a fairly narrow field, and pure-astrophysics jobs (particularly tenure-track academic ones) are relatively rare. There's definitely a lot of competition!

That said, there are few fields of study cooler than astrophysics, and with today's array of large (8-meter-plus) land-based telescopes and space telescopes, not to mention the LIGO/VIRGO/etc. gravitational-wave observatories, the number of new discoveries and the amount of new data showing up is incredible. The upcoming James Webb and WFIRST space telescopes and 30-meter-class ground-based ones will truly open the floodgates. It's a very exciting time to be in the field.

So the answer depends somewhat on how flexible you're willing to be. Physics in general provides an excellent foundation for all sorts of analytical careers; the debugging and data-analysis skills you'll learn in observational/experimental (astro)physics, the mathematical skills on the theoretical side (coursework), and, more than likely, the computational skills across the spectrum, are all hugely useful and can provide opportunities in a bunch of areas. I did computational and theoretical work on disc galaxies, decided I liked the computational part just as much as the astrophysics, and ended up in corporate research initially, followed by embedded systems, web search, high-end cluster infrastructure, and data infrastructure. Pretty much all of it's been fun and has paid well, but I have had to be willing to learn new stuff along the entire path, and my location since grad school (Silicon Valley) undoubtedly helped.

If, on the other hand, you're dead set on getting an academic position, statistically your chances aren't very good, and your quality of life along the way also may not be very good. :-/ Grad school, while frequently "free" for physics/astrophysics students (via tuition waivers and teaching-assistant or research-assistant support) pays little more than what you need for room and board, and postdocs aren't much better off (and generally have to fight for grants). You're typically looking at 5-7 years for grad school and 4-8 years as a postdoc, which implies more than a decade of near-poverty before you even get the opportunity for a tenure-track position. This article will give you some idea of what grad students face, but the overall career outlook is described in this one--and it's not pretty. :-( While the problems are now being recognized, it remains to be seen whether they'll be addressed in any substantive way, and how quickly.

But to end on a high note, the work in physics/astrophysics academia is "stimulating and meaningful," and that may outweigh all other considerations for you. Certainly you shouldn't shy away from an undergraduate degree in astrophysics; you'll still have many options open to you when you graduate, in both academia and industry.

That is a very good question? From my years in Human Resources, I found that your success in the pursuit of a position in any career area all depends upon how diligent you are in your search for answers. Talk to your favorite teachers and follow their suggestions about meeting and talking to people who are working in that field who can give you some helpful information. Talk to the person who tracks and works with graduates of your school to locate and talk to and meet and visit graduates who might be working in this area. Talk to anyone that you can and ask them who might be good people seek out who might be able to provide answers to how one such as yourself might be able to enter into that career field.