Can multiple electron pairs be present in a single orbital? An electron pair as a composite system acts as a boson. So why can’t three/four electron “pairs” be present in a single 1 s Hydrogen atomic orbital?
 A: This is actually a good question, because there are composite particles that we treat as bosons. A great example is the atoms in magneto-optical traps that are cooled to produce Bose-Einstein condensates (BEC). All the constituent particles are fermions (electrons, neutrons, protons), but the composite particles are bosons. 
The difference here is that the composite bosons (the atoms) are small compared to the size of the trap. The density of the resulting BEC is not actually higher than the density of an ordinary liquid, so we are not trying to squeeze the electrons themselves into a single state. 
In the case of electrons in an atom, they are bound to the nucleus, not each other. The singlet of two electrons in the 1s orbital in Hydrogen aren't a single object $(e_{\uparrow}  e_{\downarrow})$ orbiting the nucleus. If you were to try to add a new electron to that state it would see both other electrons and want to form an antisymmetric state with both of them, which is cannot, therefore it can't be in the orbital. 
References: 


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*See the "quantum states" section Bosons. 

*Pauli Exclusion Principle 
A: 
An electron pair as a composite system acts as a boson.

Hooold up.
If by "composite system" you mean a bound system, like a Cooper pair, then yes. In this case, two electrons are bound together so that you can stop talking about the individual electrons (two fermions) and instead describe this new state as a new "particle". Which has integer spin. So it's a boson.
But if by "composite" you just mean that I can draw a box around them and the two electron happen to fall within it, then no. If they still go on and about on their own accord then they will be described by their individual wavefunction and they all all behave like fermions. In an atom, for example, the electrons still follow the Pauli exclusion principle and pile up in ever higher energy levels. They actually do interact among each other (e.g. via Coulomb repulsion) but not in a way, like for the Cooper pair, to be described by a new excitation.
