Set theory, and in general logic, has a big portion devoted to "how far can we stretch this theory?" and since inconsistencies in FOL appear instantly, there is no "a little bit of contradiction".
What do I mean? Well, we work in a certain framework, this framework has rules of inferences which do not allow "a little bit" of inconsistency, but rather explode once inconsistencies are introduced.
Godel's theorem shows that if our theory is strong enough then if we can effectively decide which axiom is in the theory, the theory will not be complete. In particular it will not be able to prove its own consistency.
ZFC is such theory, it has the properties needed for Godel's incompleteness theorem, and in particular has the capabilities of expressing its own consistency. Therefore it cannot prove it. Indeed we always have to assume that ZFC is consistent.
Large cardinals are additional axioms, those are strong axioms which often prove the consistency of ZFC. This is quite a leap of faith, so to speak, and we always have to be wary that we do not introduce contradictions. Contradictions are of course assumptions from which we can prove everything, in particular the consistency of ZFC.
When we add the axiom "There exists an inaccessible cardinal" we may have added an inconsistency, but we did not find such yet (modulo some claims which are unverified). This is a very mild form of large cardinal, but it is an additional axiom nonetheless.
We can add more, two inaccessible. Infinitely many inaccessible cardinals, or even a proper class thereof. Those still do not yield a clear contradiction and thus we are still okay with believing those axioms are consistent.
Going even further we reach the point of measurable cardinals. There we find a subtle point which is expressed in Kunen's inconsistency's theorem. Namely, measurable cardinals are critical points of elementary embeddings of the universe into a subclass.
If we assume this embedding to be onto the universe then we have derived a contradiction. This is something which we do not want, so while we cannot approximate it by "little inconsistency" or "a bit of contradiction" we have to approach from a different angle. By studying the exact reason from which we derive the contradiction we are able to write axioms which are closer to it but not strong enough to prove it.