What is the formal definition of a system?

I will stick with this definition for a physical system:

In physics, a physical system is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the system.

You ask for many body systems:

So the question is: what is the formal definition of a system? When are these fermions entangled and cannot occupy the same state and when aren't they?

As an experimentalist I have found the density matrix formalism as useful for developing intuition about many particle systems.

If the density matrix of the many body system has non zero off diagonal elements, it means that the particles in the chosen system are represented by one quantum mechanical wavefunction. The off diagonal elements show the degree of "entanglement" for the particular pair of particles.

This question and its answer might interest you for neutron stars.

Two identical fermions cannot be in the same state, period.

Location is part of the state, so two fermions in different locations are automatically not in the same state. "Gaussian wavepacket at $p$ momentum at the LHC" and "Gaussian wavepacket at $p$ momentum at LANL" are different states.

In principle all identical fermions have an exchange interaction regardless of distance, but in practice it's only significant at short distances.