Yes, it is completely possible.
As a matter of fact, there are several different kinds of software that already exist that do this. Though I know several more exist here is a list of the ones I am aware of:
I am only familiar with older versions of Hamachi as I have not used it often since the software was acquired by LogMeIn.
Because there are so many solutions, I recommend that you start with those I have mentioned as well as looking for others and see if you can find the one you would prefer. At one point there were several websites dedicated to just showing these networks and what games were being played on them. I dont have any information on those sites anymore but I am sure there are others here who might have some knowledge.
As a side note, I have heard of issue where some older games would not work with this kind of software though I would hope others have found ways to fix this now.