If I understand its formulation correctly, the completeness theorem states that any consistent theory has set-sized models, as opposed to models of any size. On the other hand, I've seen a lot of answers phrased something like this:
It's consistent with ZFC that there are no set models of ZFC, but there would still be class models such as the von Neumann Universe.
Which seems a bit strange, because it seems to suggest that ZFC could have only proper classes for models, despite being consistent.
I realize this is partially about which things are sets in which models: if a smallest inaccessible cardinal $\kappa$ exists, then the model $V_\kappa$ is a model of ZFC and none of the sets in that model are themselves models of ZFC (right?). But it still seems like we should be able to talk about whether set models "actually" exist, in terms of something like: "There exists a model of ZFC one of whose elements is an inner model of ZFC."
So the question is: what am I missing in the assertion that ZFC is relatively consistent with the absence of set models? Does this just mean that some models of ZFC have no elements which are inner models? Or is it saying something stronger along the lines of "it's possible that although ZFC is consistent, none of its models can be formalized as sets in any other model of ZFC?"