Oooooh.

Now, this is going to be a kind of longish post.

In the field of maths, a mathematician by the name of Kurt Godel came up with a very elegant proof of a very interesting idea - no matter how powerful a system one derives to describe mathematics, that system will contain a statement which is unprovable, even though it is true. This led to a related bit of work by Alan Turing, who proved the existence of what are called undecidable problems - problems where there is no definable solution in the context of a machine. I'll skip the details - for a good explanation, look for 'halting problem' and 'Church-Turing thesis'.

The bit where this starts to get -really- interesting (and I get on-topic), is that the machines that Turing worked with are all of the same type in terms of the set of problems they can solve. In fact, we don't actually know of any practical ways of computing things that are not equivalent to these machines, that is to say, of solving problems outside of this set. It's even an open question as to whether we, in terms of our brains, aren't equivalent to these machines (in the sense that they can do anything that we can do ignoring speed differences).

I'd argue that there -might- exist problems that are 'beyond the realm of scientific enquiry' by way of this argument.

1) Assuming that our brains are equivalent to these machines by way of the Church-Turing thesis;

2) There exist problems where there is no definable solution in the context of these 'machines';

3) Therefore, there exist problems where there is no definable solution in the context of our brains;

4) Science is defined as being (per wikipedia) "in its broadest sense, any systematic knowledge-base or prescriptive practice that is capable of resulting in a prediction or predictable type of outcome";

5) Science is a construct of human logic and reasoning.

Therefore;

There exist questions/problems that are beyond the realm of scientific enquiry because of the fact that there are problems where there is no definable solution or outcome when we apply our brains to them, and science is a systematic approach capable of predicting outcomes, created by humans.

The reason for the 'might' is simply that this argument does hinge on human brains/logic being equivalent to these machines. This is, as I said, an open question. It also doesn't really help to give you an example of such a question.