People who specialize in psychology can tell you the answer to this easily: yes, no question about it.
Psychology actually has a strong base in biology and many mental illnesses have theorized neurological causes. These things are very observable. To not be a science, psychology would have to study something that one cannot observe or measure. But you can and psychologists have to use the scientific method like any other scientist. Psychologist use systematic methods to study the mind, which includes both subjective and empirical observations. Some of them are less "scientific" than others, like surveys, interviews, questionnaires...but these are only in certain fields of psychology like behavioral psychology or social psychology.
Yes, it is studying something that we meager humans will never be able to fully comprehend (nor would we really want to); but so is many other forms of science. In this natural world, in this big universe, we humans can't really understand everything or find everything. It's foolish to assume you can. All science is limited by this fact...the fact that we will never be able to really know everything about what we are studying. So psychology is no different from astronomy, zoology, etc.
And yes, psychology as a pretty big margin for error seeing as though when observing mental illnesses, you're relying on a lot of subjection from both yourself and the person you're interviewing (though that's only in certain branches of psychology). But again, so does every other science. Can you really be sure that the equation for gravity is really Fg = G*m1*m2/r^2? Can you really be sure that an atom is the smallest molecule? Can you really be sure that the "planet" you're observing thousands of lightyears away is really there? The answer is no. These are all observations based on a gathering of data that can easily be proven wrong at any point. A big part of science is realizing that nothing you may come up with is truly "fact", but merely theoretical.
So, if we are to assume that psychology is not a science because one assumes its subjective, theoretical, and studying a subject that will never be fully understood...then we would have to assume that there simply is no such thing as science. If something is a science because it uses the scientific method to gather empirical data and it studies a tangible, observable subject...then how is psychology NOT a science?