The biggest obstacle is that you haven't been through a Ph.D. program. Not (just) because of formal qualifications, but also because a Ph.D. program is where one learns how to manage an independent research program. The reason Ph.D. programs are a de facto requirement for research positions is that very few people learn how to do this in a bachelor's or master's program. It's not impossible, but it's certainly rare. (It's already difficult for recent Ph.D.s to get research grants in competition against much more experienced researchers.)
Aside from appropriate research experience, what you need is a formal affiliation with a university. In the U.S., it's called a "soft-money position." This is a position paid for entirely by research grants, without salary or funding from the university. (Soft money is money that depends on outside grants, while hard money is budgeted from the university itself.) If you can get the grants in the first place, it's much easier to get a soft-money position than a regular job, since there's no risk for the university: as long as your grants continue, they can collect overhead to pay for office space, computer and library access, etc., but if your grants end then so does your job. Of course nobody will give you a soft-money position unless they are impressed with your work and think you would be valuable to have around (and would not hurt the department's reputation), but this is a much lower bar than convincing them to spend their own money on you.
A soft-money position is the closest thing I'm aware of to applying for funding as an independent researcher. (It's not completely independent, but about as close as you are likely to come to independence.) However, in the U.S. it would be nearly impossible to get such a position with just a master's degree.
Whether this path is feasible at all depends on your research area, and of course your funding agency's policies. In the U.S. it's pretty common in medical research, but almost unheard of in mathematics (where there is much less funding available). In computer science it's somewhere in between, depending on the specific subfield.