Either consciousness lives in the physical constructs of the human brain or it is supernatural / "magic". If the first case is true, then there is every reason to believe it is open to the scrutiny of the scientific process. What we know for sure is that there are very many scientific questions about the brain that still remain to be answered before we have to throw up our hands and admit that it is all ineffable.

IMO science is doomed to always leave something out as you say.. it will reach a fundamental building block that simply exists, can not be expressed in terms of something else and might as well be called God. But i just haven't heard any convincing arguments that consciousness can't be understood before we reach that point.