Why 2 electrons can't be in the same quantum state when they are distant apart? I understand that when 2 electrons are confined into a very small volume of space slightly bigger than their debroglie wavelength, one of the pair must jiggle with increase momentum due to pauli exclusion principle.
But looking at G. Smith's comment in my earlier question, why can't 2 electrons separated with a vast distance of space share the same quantum state? It didn't make sense to me unless the electrons are bound to an atom then each of them must go to different energy level since already 2 electrons with the same lowest energy state show different spin state. 
 A: Electrons, in general, don't necessarily have well-defined positions, so the idea that they are "separated with a vast distance of space" is nebulous, at best. I'm going to assume that when you say 

2 electrons separated with a vast distance of space 

you mean 

2 electrons whose wavefunctions are well-localized (i.e. strongly peaked in position space) and for which the peaks of the wavefunctions are separated by a large distance.

As you can plainly see from this characterization, the wavefunctions of the two electrons are very different, since they peak in different places. The wavefunction is part of the electron's quantum state.* This means that, since the electrons have different position wavefunctions, they are in different quantum states.

*There are other ways to define the quantum state that don't directly reference position, such as giving a momentum-space wavefunction or a decomposition into eigenstates of a particular potential, but the same principle applies - if whatever you use to represent the quantum state is different for one electron than for another, they're in different quantum states.
A: I thought I'd add one additional comment based on the comment from G Smith.  If there were a box that was a billion meters across, all of the quantum eigenstates would be delocalized over the entire box, so even though the box was really big, the electrons in the same states wouldn't actually be separated.  Now, you could certainly make linear superpositions of those functions create states localized on each end of the box, but then they clearly wouldn't be in the same state to begin with.  
A: The Pauli Exclusion Principle is equivalent to the statement that any multi-electron wavefunction must be totally antisymmetric because electrons are identical fermions. This antisymmetry  requirement holds regardless of the size of the domain of the wavefunction  (e.g., it can be defined on infinite Euclidean space) and the number of electrons. So you can have two electrons in infinite space and they can’t be in the same state, because otherwise the wavefunction would vanish, even though the number density of electrons is zero.
