Irrationality of numbers in ZFC Define the number $\alpha$ as $\alpha=\sum _{n\in\mathbb{N}}\frac{f(n)}{10^{n!}}$, where $f(n)=1$ if there's a proof of 0=1 in ZFC of length n, and $f(n)=0$ otherwise.
Then, the statement "$\alpha$ is a rational number" (or an algebraic number for that matter) is independent of the axioms of ZFC.
However, that example is very contrived.
Could it be possible that the statement "$\alpha$ is rational" is independent of ZFC, where $\alpha$ is a more familiar number such as $\alpha=e+\pi$?
 A: In principle, sure! In practice, we have no reason to believe that such a statement is independent of ZFC.
Note that the sentence "$e+\pi$ is rational" (say) is a relatively simple one - in the hierarchy of complexity of sentences (see e.g. here), it is at the $\Pi^0_2$ level. Such statements cannot be proved independent via forcing (which is currently the most prolific way of demonstrating independence from ZFC), or via inner models (which is the second most); so the only obvious way to show that it is independent of ZFC would be to show that it is related to the consistency of ZFC. The problem is that such arguments generally go through fast-growing functions, and there isn't obviously a fast-growing function associated to the (ir)rationality of $e+\pi$.
Now this doesn't mean that "$e+\pi$ is rational" is decidable in ZFC, of course! For one thing, there are extremely clever arguments for independence which are more subtle than what I've outlined above, and which I know nothing about; Harvey Friedman has done a lot of work in this area. Conceivably such a technique could apply here. And of course even if no currently known technique suffices, that doesn't mean we won't find one later. And - worst of all! - it might be the case that we can't prove that "$e+\pi$ is rational" is independent of ZFC, even if it is! 
So all the evidence we have in any direction is entirely heuristic. Personally, I believe "$e+\pi$ is rational" is decidable (disprovable, specifically) from ZFC; but I have to admit that I have no real evidence for this belief.
