It is always weird to see a proof that something is impossible, especially when the tools used in the proof have nothing to do(at a first sight) with the original statement of the problem. I know a few such mathematical theorems, which prove that some facts cannot happen in various areas of mathematics like geometry, algebra, analysis. One example is that there isn't a general formula, using only radicals, for solving polynomial equations of degree greater than $5$. The proof of this theorem uses Galois theory, and this tool was invented especially for solving this problem.
What other theorems do you know which prove that certain facts are impossible in mathematics, and in their proofs something unexpected is used?
Since I was asked to be more specific, I'll tell you more about what I was thinking. I see that a first answer, about the impossibility of the trisection of the angle was posted. This is exactly the kind of answers I have in mind. It is proved that an angle cannot be split into three equal parts using a straightedge and a compass, but the proof has nothing to do with geometry. It uses Galois theory.
Another example: It is impossible to dissect the unit square into an odd number of triangles with equal areas. This is known as Monsky's Theorem, and the only known proof(as far as I know) uses $p$-adic valuations arguments, again, something that we normally wouldn't expect.
I found myself some examples:
Impossibility of making some construction with ruler and compass.
Impossibility of calculating $\int e^{x^2}dx$ in a reasonable closed form. ( I first heard of this one when I was in my first year of college, but the teacher didn't say who proved it, or how it can be done.)
Best Answer
A natural impossibility question in knot theory (perhaps the most basic and important one) is to ask whether it is possible, given two knot diagrams, to transform one of them into the other by means of the Reidemeister moves. By Reidemeister's theorem, this is equivalent to them presenting the same knot. Of course if it is possible then we can just exhibit a sequence of moves, but if it is impossible, how to prove this is not so clear.
One method is to construct knot invariants. These are objects such as polynomials constructed from knot diagrams and invariant under the Reidemeister moves; thus, if two knot diagrams have different invariants, they cannot present the same knot. A simple example is tricolorability, which already proves that the trefoil knot is not the unknot. More sophisticated and powerful examples such as the Alexander polynomial and the knot group can be constructed using topological methods (but using the knot group is not as easy as using polynomials because it is undecidable in general whether two groups are isomorphic given only their presentations whereas it is fairly straightforward to decide whether two polynomials are equal).
In 1984 Vaughan Jones discovered the Jones polynomial in a fairly unexpected way while studying operator algebras. This discovery pioneered the now very active field of quantum knot invariants, which has ties to many other fields of mathematics. I don't know a great introductory survey on these ideas but you can consult, for example, Turaev's Quantum invariants of knots and 3-manifolds, and there are also some good keywords and links at the nLab page about knot invariants.