Where do models of set theory live? When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of $ZFC$ or extensions of it like $ZFC+CH$. 
So when we say, for example, that $CH$ is independent of $ZFC$ we are saying that there exists a model $M$ for the theory $ZFC+CH$ and another model $M'$ for the theory $ZFC+¬CH$. That models cannot be objects of ZFC since in that case $ZFC\vdash Cons(ZFC)$ so, where they live? 
I feel that there is a formal notion that I'm ommiting or misunderstanding so, could anyone give some explanation about it?
Thanks in advance.
 A: It is not really meaningful to say that such and such "models cannot be objects of ZFC", because there is no such thing as "objects of ZFC". What there is are objects of particular interpretations of (the language of) ZFC, but different interpretations of the same theory may or may not contain objects with particular properties.
(Some of these interpretations are models, when their universe is a set in whichever set theory we use for our metareasoning. If we believe in a Platonic universe of "actually existing" sets, that may also work as an interpretation of ZFC, but it usually won't be a "model", depending on the exact definitions we employ).
This distinction is important here, because you appear to be confused about the difference between some object existing in some particular interpretation, and the theory proving they exist.
If ZFC is consistent, then there will be some interpretations of it that contain sets that are models of ZFC itself, and others that don't. In particular every model of ZFC whose integers are standard will contain a set that is a model of ZFC.
On the other hand, the Gödel-Rosser incompleteness theorem says that if ZFC is consistent, then it cannot prove its own consistency, and therefore it cannot be the case that every model of ZFC contains an smaller model of ZFC. Those that don't, however, cannot have the same integers as we use outside the model (because the non-existence of an internal model of ZFC implies that there is a number that the outer model believes is a proof of a contradiction from ZFC, and such a number cannot be standard).
