Pauli Exclusion Principle and Quantum States We know that two identical fermions cannot be in the same state together because of the Pauli exclusion principle.
My questions are:


*

*Can two bosons (for example, photons) be arbitrarily close together? Can we keep an infinite amount of photons in a box as small as possible? Or, perhaps it actually means, two bosons can just be 'under' the same wavelength?

*Does "states" here mean both spatial and energy states? If two electrons with the same spin direction have different amount of energy/momentum, therefore they are already in different quantum states, can they be in the same spatial state, meaning the same location? Here we are assuming free electrons, not electrons in an atom, so I guess discussion of the atomic orbitals might be irrelevant.

*How close can two electrons with the same spin and same energy state be together? Is it related to their de Broglie wavelengths?
In short, exactly what is the meaning of the 'state' in Pauli Exclusion Principle, and how close does two identical fermions need to be to be in the same spatial state?
 A: 
exactly what is the meaning of the 'state' in Pauli Exclusion Principle?

The most simple and complete answer is: the Pauli exclusion principle refers to eigenstates. Your questions seem to refer more to a particle view of matter. "How close" is a question that we can ask about particles, not about quantum states.
So, let us consider the questions from the point of view of quantum mechanics, that is, considering the waves resulting from Schroedinger's equation.

Can we keep an infinite amount of photons in a box as small as possible? Or, perhaps it actually means, two bosons can just be 'under' the same wavelength?

1) You can put as many bosons in a box. You can put as many photons in a box, with exactly the same wave length. It is what you do when you make a laser. Of course, people must be smart to do so, but it is feasible. In this case we are speaking of eigenfunctions of the electromagnetic waves. Each of them corresponds to an harmonic oscillator. Once quantized, the eigenfunctions will correspond to solutions with $n$ photons (with arbitrary $n$) so, yes, we can put as many photons in a box as we want.

Can two bosons (for example, photons) be arbitrarily close together? 

Here we should define what it means "close" for waves. Here you are not asking about eigenfunctions of the electromagnetic field, but rather wave packets. Also in this case, nothing prevents to superpose as many photons as you want. Of course, also in this case there can be practical difficulties, e.g. if you have a too intense beam through a medium you can have non-linear phenomena.

Does "states" here mean both spatial and energy states?

2) "State" means "quantum state". Same eigenfunction. So, same expectation values for energy, momentum and anything else. But do not mix this with the particle interpretation. Two bosons in the very same quantum state (eigenfunction), when detected, can show different properties, because of the stochastic nature of quantum phenomena.

How close can two electrons with the same spin and same energy state be together? Is it related to their de Broglie wavelengths?

3) Two electrons cannot stay together: they repel due to electrostatic interactions. I can interpret the question as: "what is the probability to find two electrons at a given distance e.g. in a helium atom?" Then, no, it is not equal to De Broglie wave length.
However, the wave length comes into play. Let us consider a 1-dimensional motion in a linear box with length $L$. Roughly, the eigenfunctions will contain an integer number of waves. Your 1d box already contains a particle, on an eigenfunction with $n$ waves. Its wavelength is thus $\lambda=L/n$. You can add another particle on a different eigenfunction with a different number of waves. The most similar will have $n+1$ or $n-1$ waves. The respective wavelengths will be $L/(n+1)$ and $L/(n-1)$. For $n\gg 1$, the wavelength difference is approximately:
$$\delta \lambda=\frac{L}{n^2} \tag{1}$$
This is the minimum De Broglie wave length difference between the particles you put in the box, which also depends on the size of the box itself. Eq. 1 can be rewritten as:
$$\delta \lambda=\frac{\lambda^2}{L} \tag{2}$$
Once $\lambda$ is fixed, the smaller is the box, the larger is the minimum wavelength difference. In turn, this wavelength difference translates into a minimum energy difference between eigenfunctions, which depends on the size of the box. This can be thought of as the requested relation between energy difference and size.
Finally, it is worth pointing out that in a 3-dimensional box we change the eigenfunction not only by changing the energy, but also by changing the wavevector (very roughly: the "direction"). Similar reasonings can be applied to the minimum wavevector difference.
