I think that for most projects there is little hope unless you get in to a doctoral (or maybe masters) program. The sort of work you envision, is mostly done by students and forms an integral part of their studies. Many projects in the sciences consist of testing hypotheses (made by students) by building things, again by those same students.
But if you are near some monster sized national or international lab in which a large number of specialists from different fields collaborate on things, then you might find a position. But we re speaking of CERN level projects, or TOKAMAK sorts of things. Or things related to astronomy or space exploration. These aren't typical projects in scope. I also can't predict what sort of career you could have doing such things as experiments eventually finish and people go on to do new things. But these are the sorts of things that require government (and grant) funding.
But if you want to get your feet wet, go to some local (large) university and ask there whether they can use some help, paid or not. This could help you decide if this is an avenue you really want to explore and also make the contacts and get the experience that could get you into some degree program. Having a doctorate is the normal path to an academic career. There are few other doors.
And if you can make a few contacts, it isn't impossible that a position might be written in for you in the next large grant proposal. But, that, again, is only a temporary solution. Grants end, though some are renewed.
However, there are also industrial labs, say IBM or Google, where the research isn't a lot different from that done in university labs. It is more applied at the current time (product focused) rather than pure theoretical research, but the organization of some of them is similar to that which you find in academia. Not all such places are very supportive, of course (think Dilbert), so you need to do some explorations before joining them. Some of the pharmaceutical companies still do real science.
And it would be a big plus in many projects if you are skilled in statistics and, maybe, mathematical modeling.