Is science objective even in principle? For one it employs methodological naturalism, which implies that it can only evaluate phenomena within a certain scope. Not that I endorse this view, but what if there are entities out there which are not naturalistic? If there were, science would be unable to study them. Plus there is the matter than science does not have unhindered access to all aspects of reality. That appears to challenge the notion that science is objective. Then there is the practical aspect, that science done by humans comes packaged with human biases. Increasingly, science is becoming a corporate matter which is focused upon applied technology for profit rather than basic research. So what often happens is that scientific findings are framed within the goals of those who are funding the research, which violates the requirement for disinterestedness. Thus, although this is an oversimplification, it appears that science cannot be objective in practice, nor even in theory (unless metaphysical naturalism is true and science has access to all possible aspects of reality). Still, none of this shows that science is not a trustworthy source of knowledge... just that it is not completely objective.
So then, how does one investigate these "not naturalistc" phenomena? Do we leave it to those knuckleheads on Ghost Adventures? Do we just accept the words of the mystics and other superstitious people? I doubt that science has access to "all possible aspects of reality," but what is the alternative? I can't think of any relevant means of investigating the metaphysical. Also, I don't believe that there is anything that is "not naturalistic." If it exists, it is natural.
Think about it. Absent tools like telescopes, scanning electron microscopes, particle accelerators, gas chromatographs and a plethora of others, our ability to investigate the world would be seriously limited (as it was prior to the invention of those instruments). We are limited to our five senses, but we have amplified them tens of orders of magnitude to reach our current understanding of the world.
I have to believe that science will continue to develop means to investigate deeper into the complexities of the universe, perhaps even discovering things we now consider to lie in the metaphysical realm.
It seems like you want to give credence to the ramblings of people who make up things like ghosts and aliens, even though their beliefs are baseless. I'm not down with that.
BTW, did you know that no one knew what causes earthquakes until after the 1906 San Francisco earthquake?