Let me reassure you -- your understanding of the Pauli Exclusion Principle is not quite correct. Two electrons at opposite ends of the solar system are quite free to have the same energy. The Pauli Exclusion Principle applies only to stationary states of electrons, or to electrons bound within the same atom, molecule, or neighbourhood. What the Pauli Exclusion Principle is REALLY saying, in practical terms, is that you can only have a maximum of 2 electrons in the same place at the same time, and those two must have opposite spin. Electrons are in the family of particles known as fermions, which obey a set of rules known as Fermi-Dirac Statistics.

The Pauli Exclusion Principle is usually applied in the context of atomic structure or chemical bonding in molecules, and solutions to the Time-Independent Schrödinger Equation. In the context of atomic structure it helps to explain the capacity of the various electron shells and sub-shells. An electron in an atom, in the single-electron wavefunction approximation, is assigned a set of 4 quantum numbers:

**n** which is the main contributor to the energy of the electron in an atom, and to its average distance from the nucleus. It is a positive integer.

**l** which determines the shape of a wavefunction. It is also an integer, and may range from **0** (spherical) to **n-1**, and can also affect the energy of a multiple electron atom.

**m**_{l} determines the orientation of the wavefunction. It takes integer values between **–l** and **+l** .It does not affect the energy.

**m**_{s} takes a value of either **–**^{1}/_{2} or **+**^{1}/_{2}.

So, to take an example, if we think of a sodium atom which has 11 electrons, each of those electrons in the atom has to have a different set of quantum numbers.

The first two go to the lowest energy shell: with quantum number sets (1,0,0,–) and (1,0,0,+). There are no other possibilities for the lowest energy shell. The next two go to a higher energy subshell with quantum number sets (2,0,0,–) and (2,0,0,+), and then six more can be accommodated in a slightly higher energy subshell (2,1,–1,–), (2,1,–1,+), (2,1,0,–), (2,1,0,+), (2,1,1,–), and (2,1,1,+). In an isolated atom these six all have the same energy. Finally, the 11th electron has no more possibilities with n=2, so it must go to the higher energy shell with n=3, where it sits all alone with the quantum number set (3,0,0,–). That is why sodium loses an electron so readily, and why the chemistry of sodium is dominated by the positive sodium ion with only 10 electrons.

There is a problem, an unresolved one, relating to quantum particles and faster-than-light communication. It comes under the umbrella of "quantum entanglement", and there is a wikipedia article that will provide a reasonable introduction if you want to follow that up.