What can one say about an inconsistent theory $T$ which has no contradictions (i.e. deductions of $P \wedge \neg P$) of length shorter than $n$, where $n$ is some huge number?

There have been some discussions about the consistency of ZFC, for instance, where it has been asserted that it would be OK for ZFC to be inconsistent as long the contradiction was enormous. However, this still seems like a bad situation to me since there are constructions which depend on consistency but don't care about deduction per se. For example, constructing a model of a theory.

Can someone explain the consequences of this?

What can one say about an inconsistent theory $T$ which has no contradictions (i.e. deductions of $P \wedge \neg P$) of length shorter than $n$, where $n$ is some huge number?

There have been some discussions about the consistency of ZFC, for instance, where it has been asserted that it would be OK for ZFC to be inconsistent as long the contradiction was enormous. However, this still seems like a bad situation to me since there are constructions which depend on consistency but don't care about deduction per se. For example, constructing a model of a theory.

Can someone explain the consequences of this?

EDIT: After hearing some good feedback, I think I can phrase this question in a more concrete way:

To what extent, and in what situations, is it possible to work consistently with a theory that is inconsistent?