It's a very interesting subject but there's nowhere in the world that it's a "safe" career option (I know archaeologists on three continents). Despite what you might be told at university, if you do go on to study it, progression is slow and the lower rung pay is terrible. After that there is a very, very tight bottleneck. A lot of people leave the profession because as they get older their financial commitments increase and they can't afford to stay. The ones who stay around after 25 hoping for an opening onto a higher career rung almost always have the financial support of their parents or a partner. I'm just laying that out there bluntly right now.
Archaeology is actually quite easy to get into as a hobby. Google around and there's a good chance that your local park authority or other local body has volunteer archaeology work. If not there are often volunteer societies that do very high quality work, also many commercial archaeology units run volunteer projects, as do some colleges. Colleges may even do evening classes in it.
Nowadays it's not as easy to get into the professional side without an undergraduate degree. How that's structured will depend on where you live. I would not advise doing a post graduate degree until after you've worked for a year or three to really find out what you're good at and where there are viable career niches. At this point you may also wish to step sideways into conservation, museum studies, graphic design, cartography, or GIS. Only in France would you need to do a postgrad directly as archaeology itself is a postgraduate specialty (or was the last time I checked).
There's a real disconnect between how it's taught and how it's practiced commercially just about everywhere. Every archaeologist I know rants about it as each year's fresh blood comes out with very unrealistic expectations. College/uni make it seem like you will specialise in an era, but in reality you specialise in a region and you will have to learn everything about that region since the last time it was scoured by a glacier to understand what you're doing. The conceptual framework you learn at uni won't be useful to you for potentially 5 - 7 years if you progress and that will be very frustrating.
One of the best ways to get progression is actually to enter archaeology obliquely via engineering (the hands-on kind), geology or soil science. They also pay better. Get your ducks in a row before trying to do archaeology full time or a life of student debt and constant scrambling will await you.