We have to hedge quite a bit in what you said. First, there are many complete theories: the theory of all groups of order 7 is complete, as an example. However, Gödel showed that any "nice enough" (read: "recursively enumerable") consistent first-order theory capable of encoding basic arithmetic is incomplete. (There is a complete theory of number theory, usually denoted $\mathrm{Th}(\mathbb{N})$, which consists of all sentences that are true in the standard model of number theory; unfortunately, this is not a "nice enough" theory — it's impossible to program a computer to print out all and only the axioms of this theory!)
Also, these statements that you gave, $\mathsf{(G)CH}$, $\mathsf{AC}$, are not provably unprovable in $\mathsf{ZF}$. However, it is provable that they are unprovable in $\mathsf{ZF}$ only if $\mathsf{ZF}$ is itself inconsistent.
This is what a relative consistency result means: it is a result that says the consistency of one theory implies the consistency of the other. Kurt Gödel proved that if $\mathsf{ZF}$ is consistent, then so if $\mathsf{ZFC}+\mathbf{V}=\mathbf{L}$ (and also that $\mathsf{ZFC}+\mathbf{V}=\mathbf{L}$ implies $\mathsf{GCH}$). Paul Cohen proved that if $\mathsf{ZF}$ is consistent, then both $\mathsf{ZF}+\neg\mathsf{AC}$ and $\mathsf{ZFC}+\neg\mathsf{CH}$ are consistent. More elementarily, with $\mathsf{ZF}$ one can construct the natural numbers, and so the consistency of $\mathsf{ZF}$ implies the consistency of $\mathsf{PA}$. These are fundamentally relative consistency results, and cannot be improved to straight consistency results. This stems from Gödel's Second Incompleteness Theorem. Because of this the acceptance of the consistency of $\mathsf{ZF(C)}$ is an article of faith (though one I'm happy to believe in).
I recommend that you look at the first page of George Boolos's Gödel's second incompleteness theorem explained in words of one syllable (Mind, vol.103, pp.1-3). It is a quite entertaining look at the meaning of Gödel's Second Incompleteness Theorem.