Is it worth it to self-fund for research experience post undergrad? Is it possible?
Labor laws are changing, particularly in the United States, and many are interpreting what you would be proposing as an unpaid internship, which is becoming harder for schools to "get away with." For example, if I want to bring someone into my lab to do research, they are supposed to be either paid or receive course credit for their work. But what I can't do for potential grad students is let them work for extended periods of months on a research problem and not compensate them for their work. 

So you may find that many people who would be interested in such an arrangement in principle may be hamstrung by their universities' policies.