I don't want this to seem like a job posting, but I'm a well travelled and able bodied male who doesn't mind manual labor or mule work. I've also had military training in the Marine Corps. What are the best avenues of approach for me to land a job/opportunity like this? I say opportunity because I'm not trying to get paid either. I really just want the experience.
What sort of expedition are you talking about? A trip to Everest or the South Pole, or are we talking an archaeological excavation?
Can't help with the former, but if you are interested in the latter, there are TONS of field schools that are looking for volunteer labor for archaeological work, many of which are overseas. I personally have been working on one excavation in Israel (Tel Ashkelon, just north of the Gaza Strip) for about eight years, and just got back from the latest trip a week ago.
There's no experience necessary for an excavation, and all it will cost you is room and board and airfare. For a dig in Israel, expect about $1500 for three weeks, and about $3000 for six weeks room and board in a hotel, and perhaps some tours on weekends. There are lots of digs, and I can recommend some if you tell me what you are looking for and what you are interested in excavating. Most digs work in the summer months, because it is vacation and they rely on volunteer labor from students and the like, and also to avoid the rain. However, there are a few digs run by universities on the quarter system that dig over the January break.
If you want to learn what life on a dig in Israel is like, you can check out the blog run by our team, http://digashkelon.com/blog, which we ran during our season, updating every day with pictures of the work we do and explanations about the things we are finding. You are more than welcome to come dig with us next summer, there will be links to applications and more infos and stuff around January, I think. Our dig usually starts the first full week of June, and goes for six weeks.
It's rather difficult to find gainful employment doing this unless you've had a field school (Even Indiana Jones had a day job). Even with lots of experience, I have never been paid to dig. I have enough experience that the dig pays my room board and airfare, so I come out even at the end of it (except for beer money), but they don't pay me, and I've got a decade of experience. If you want to get paid, I recommend looking into a local university and taking a field school, it may require some reading and stuff. Then you can explore places like http://www.shovelbums.org/ where they advertise for archaeology jobs in the States, and you can apply for labor positions. But they pretty much all require at least a field school and some require an undergrad degree.
That's the basics, and I can give you much more info if you have an idea of what sort of expedition you want, and if it's archaeology, where you would be interested in digging.
I think you'll have better luck in /r/iwantout or the Peace corps
Did you mean to post this to r/askhistorians?