The type of questions that I will discuss here are the deep questions about manifest existence, such as "What is the universe?", "What is the nature of reality?", "What is consciousness?", "What are qualia?", "What is the self?", "What is space / time / matter / energy?", and so on.
Whether science can potentially answer these or not is a subtle issue that deserves a detailed answer.
Firstly, science is not a static and monolithic entity, it is quite diverse and it is constantly evolving. Furthermore, not all areas of science are in agreement or even compatible; i.e. there exist profound contradictions within science. These are part of the creative tension that drives its evolution through successive paradigm shifts.
A paradigm shift changes not only the axioms and resulting narrative of science, but potentially also its epistemological foundation as well as the principles of its practice and its goals.
Furthermore science is conducted by socially conditioned people, thus cultural shifts have a profound impact on what 'science' means to scientists and how they enact scientific practice in their day-to-day work.
I will here argue that the inability of science to address the deep questions of manifest existence is not an inherent limitation of science in general, it is merely a limitation of all forms of science that have currently existed.
However some branches of science are getting very close to breaking through to a new paradigm with which they could potentially overcome this limitation. See "Signs of an Emerging Paradigm Shift" http://bit.ly/dfxL4R
The fundamental obstacle preventing science (and most people) from coherently approaching the deep issues is the cognitive structure and functioning of their minds.
Naïve realism is a cognitive habit of apprehension (perception, interpretation and conceptualisation) that causes the mind to unquestioningly accept certain unfounded assumptions about the nature of that which is apprehended. For example, if you look at a chair and immediately believe that you are looking at a physical external chair then that is because of naïve realism. Without naïve realism all that one really knows is that one is experiencing sensory impressions and memory associations that lead one to think of the learned concept 'chair'. Anything else rests on unfounded assumptions and is thus unscientific and anti-sceptical.
Naïve realism is biologically useful; for example if one happens to see a tiger on the prowl one shouldn't philosophise about the nature of perception and cognition but instead one should run and hide. Hence naïve realism is deeply engrained within the mind. However in order to address deep philosophical issues one must overcome this instinctual cognitive habit and become fully sceptical and scientific.
Because naïve realism is a natural feature of the cognitive structure of the mind empiricism was an obvious early attempt at a scientific method.
"Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge" (Rationalism vs. Empiricism, Stanford Encyclopedia of Philosophy http://bit.ly/9JZrtj)
This is because the objects of sense perception where believed to be the whole of reality. This however overlooked the consciousness that was apprehending them and within which they existed as perceptions. Upon this empiricist epistemology we have constructed a narrative that describes a "material universe".
"materialism is the philosophy of the subject who forgets to take account of himself." (Schopenhauer)
However every attempt to discover matter (as it has traditionally been understood) has so far failed.
"Because it is now a scientifically established fact that less than 5% of the universe is composed of matter as commonly understood modern philosophical materialists attempt to extend the definition of matter to include other scientifically observable entities such as energy, forces, and the curvature of space. However this opens them to further criticism from philosophers such as Mary Midgley who suggest that the concept of "matter" is elusive and poorly defined." (http://en.wikipedia.org/wiki/Materialism)
"Let us now return to our ultimate particles and to small organizations of particles as atoms or small molecules. The old idea about them was that their individuality was based on the identity of matter in them... The new idea is that what is permanent in these ultimate particles or small aggregates is their shape and organization. The habit of everyday language deceives us and seems to require, whenever we hear the word shape or form of something, that it must be a material substratum that is required to take on a shape. Scientifically this habit goes back to Aristotle, his causa materialis and causa formalis. But when you come to the ultimate particles constituting matter, there seems to be no point in thinking of them again as consisting of some material. They are as it were, pure shape, nothing but shape; what turns up again and again in successive observations is this shape, not an individual speck of material." (Erwin Schrödinger)
In the meantime there has evolved another scientific method called rationalism. It has its roots in the emergence of mathematical physics and has reached maturity in the form of quantum mechanics. It is not an empirical science.
"Rationalists claim that there are significant ways in which our concepts and knowledge are gained independently of sense experience... there are cases where the content of our concepts or knowledge outstrips the information that sense experience can provide... reason in some form or other provides that additional information about the world." (Rationalism vs. Empiricism, Stanford Encyclopedia of Philosophy, http://bit.ly/9JZrtj)
Thus unfettered from the limitations of sensory appearances
"We now know that quantum objects behave differently from everyday objects, and we can make an experimentally supported epistemological claim about the quantum world, a very realist claim." (A Critique of the Empiricist Interpretation of Modern Physics http://bit.ly/aTYApv)
Due to the emergence of rationalism and in particular quantum mechanics, realisations are being made that would have been impossible within empirical science, for example
"We have no satisfactory reason for ascribing objective existence to physical quantities as distinguished from the numbers obtained when we make the measurements which we correlate with them... we get into a maze of contradiction as soon as we inject into quantum mechanics such concepts as carried over from the language and philosophy of our ancestors." (The Fundamental Principles of Quantum Mechanics, E. C. Kemble, McGraw Hill)
This realisation cuts to the heart of naïve realism and undermines the very reality that empiricists ascribed to the objects of sense perception. Hence it is not surprising that quantum physicists have struggled for over 80 years to assimilate the implications of quantum physics. Even today the majority maintain an attitude such as
"The one thing that can be said against it is that it makes absolutely no sense!" (Quantum Concepts in Space and Time, Roger Penrose)
There are deep and subtle paradigmatic attachments within the mind thus the mind instinctively acts in self-defence when it encounters ideas that it cannot assimilate using its current cognitive structures. In such circumstances it exhibits a behaviour described by Piaget as cognitive repression.
"Piaget has invited the comparison between the historical development of scientific thought and the cognitive development of the child. Both, it is suggested, proceed through the emergence of discrete stages of structural organization, each stage bringings with it new possibilities of conceptual integration, and concurrently, the possibility of a verbal articulation of the new level of organization perceived. Prior to the establishment of a new conceptual structure, knowledge already present in nonverbal forms (in e.g., sensorimotor rather than representation schemes) finds no avenue of expression, and, to the extent that it jars with the earlier established structures, demands cognitive repression. Piaget tells us that an action schema which "cannot be integrated into the system of conscious concepts is eliminated... (and) repressed from conscious territory before it has penetrated there in any conceptualized form." Caught in a transition between stages, the child, when pressed to articulate perceptions requiring cognitive structures which are not yet available, displays confusion, denial and avoidance - a disequilibrium strikingly reminiscent of the mechanism of affective repression."
(Cognitive repression in contemporary physics, Evelyn Fox Keller, http://dx.doi.org/10.1119/1.11911)
Cognitive repression amongst quantum physicists is weakening and there are a growing number beginning to declare that
"Quantum mechanics forces us to abandon naïve realism." (Quantum Enigma: Physics Encounters Consciousness, B. Rosenblum and F. Kuttner, http://bit.ly/bnrOAQ)
This all sounds rather mystical; there are even some declaring that
"The philosophy of mystical traditions, the perennial philosophy, is the most consistent philosophical background to modern science." (Fritjof Capra)
It just so happens that the core aim of mystic practices is to overcome the influence of naïve realism on the mind. This is so that one may apprehend reality more clearly, rather than filtered through a cognitive structure, which although biologically useful, is a radical distortion of reality. Overcoming this distortion is vital for an accurate apprehension of reality, thus it is a vital step that scientists and science as a whole must inevitably take.
As the influence of naïve realism wanes within our cultural discourses and our minds, science will eventually be able to enquire into the fundamental questions of manifest existence.
As an example of a possible approach, consider some of my answers on Quora, which in a crude way sketch out a possible narrative of a non-naïve realist science.
The issue of "What is the 'substance' of this universe?" is touched on in John Ringland's answer to What is consciousness?
The issue of "What is it that causes this universe to exist and function?" is touched on in John Ringland's answer to The Big Philosophical Questions: Is the Universe a Simulation?
The issue of "How does such a simulator work?" is touched on in John Ringland's answer to Is it possible to create a general system simulator, if so how?
The issue of "How is it that sentience can arise in a simulated virtual reality?" is touched on in John Ringland's answer to What is sentience?
The issue of "What is the structure of this universe?" is touched on in John Ringland's answer to What is a fractal?
The narrative potentially points toward a paradigm within which the deep questions of manifest existence are amenable to scientific enquiry.
I hope this sheds a little light on the issue of the limitations of science.
BTW some related questions on Quora: