Is it possible take take part in a research project if I'm not a part of a university?

It's certainly possible, though admittedly somewhat harder outside the framework of a university. Some potential avenues, answering generally - not all of them might apply to your particular circumstance.

1. Academic/Business partnerships. These are a new hot topic, and in some fields quite active. Universities love them because they're a revenue stream. Businesses like them because its harder to get closer to the cutting edge than at a major university. Look for companies that do this as potential employers? They're good for both dabbling in research, and also as a springboard into the research side of things - I've met several "private sector refugees" in my time.
2. Research-oriented companies. Quintiles, RTI, Westat, RAND, etc. all come to mind. There are tons of these companies, and many of them both pay quite well and actively publish. Are there any that serve your particular field?
3. Consulting. Research groups occasionally have funding for outside contractors of one sort or another - and if someone really wants to work with you, they may write such a position into a grant with you in mind. For example, I have some grant support for a freelance programmer. I've known other people hired for a particular expertise, or just "a warm body who isn't a student". This is probably the path you'd end up going down if you both contact your ex-professors and want to get paid.
4. Volunteering. Academics are cheap. It never hurts to ask if they've got some side project you might be suited for collecting dust in the back.

The short answer is yes, you can.

There is no need to be in a university, a research lab to work on a research project. There is also no need to have a PhD or any other specific degree/position. This is however somehow unusual. But, there is a bunch of people in industry that publish on a regular basis some serious research work.

If you have no idea of the problem you want to address, or at least a broad domain, this will be even harder. I can only advice you to offer your manpower to people who are established researchers. For that purpose, you can go to seminars, you can maybe, if this is possible where you live, lurk into some graduate lectures, etc. After a while you will be able to talk with people and offer your services. However, the simple way is to work with relatives. For instance, my brother is an entrepreneur, and he wrotes two papers with me (one is published, the other will be soon I hope). Of course, it was simple: I described one of my problem on a sunday lunch at the parents house, He had some ideas, we worked on it...

It is possible to take part in a research project, but your situation is not that common.

Many labs have paid positions for Research Assistants, where you will assist with managing all aspects of the research except for defining the problem to be solved. Details of the position will vary from lab to lab, but you will often be very involved in the research. Usually research assistants are not authors on publications, so you would need to discuss the details with the professor managing whatever lab you're interested in.

Also, many labs will take on undergraduate student volunteers to assist them with their research. Your having completed an undergraduate degree may make you eligible for these positions, on a case-by-case basis. You would want to speak with individual labs to see whether they would be amenable to this sort of arrangement. Note that position would almost certainly not be paid. Also, note additionally that there may be insurance issues which would preclude you from following this path... you'll have to look into this for yourself.