What would happen if two electrons fuse? Two electrons repel each other naturally; however, if protons can be forced to fuse, can electrons technically fuse too? What would be the product if two electrons fuse?
 A: 
if protons can be forced to fuse, can electrons technically fuse too

I'm not aware of any known mechanism by which two electrons can 'fuse'.
It is possible, however, for two electrons to form a Cooper pair which is the basis for superconductivity.
However, this cannot occur between two otherwise isolated electrons.
A: Two protons (or more) protons can stay together because of the effects of the residual strong nuclear force. It's called the "strong" force because it's extremely strong - about 100 times as strong as the electromagnetic force. Without it, protons would fly apart.
The strong force is mediated by gluons, the force's bosonic carriers. In order for electrons to interact via the strong force, they would presumably have to interact via gluons. The problem? There have been no observations of electrons interacting via gluons. In fact, if they did, they would interact with other particles that feel the strong force - such as protons and neutrons - in ways that are different from what has been observed.
So now you have to invent a whole new force, and explain why it hasn't yet been observed.
A nice summary of the strong nuclear force's role in the nucleus can be found here.

What would be the product if two electrons fuse?

Here we have to consider some conservation laws. Quantities that are relevant here that would have to be conserved are


*

*Electric charge

*Angular momentum (in the form of spin)

*Lepton number

*Linear momentum

*Energy (including mass)


In this case, you could not have one resultant particle, because that would violate conservation of lepton number. You would need more. So now you're not really "fusing" two electrons but are instead describing a complex interaction. And that doesn't even take the other laws into account.
A: The process of 'smashing' electrons on electrons is called Møller scattering. Essentially, you'll get two electrons again (but with potentially different direction). Since there is no attractive force between them, you will not get what is called a bound state, i.e. they will not stay 'close' to each other.
A: Using energy arguments, even if you get two electrons moving towards each other at (or very very near) the speed of light, the minimum distance they can reach is the Compton radius.Even an electron and a positron can't get nearer than this due to radiation losses in the process. see https://en.wikipedia.org/wiki/Classical_electron_radius
A: First of all, two isolated protons cannot fuse to form a bound state. The nuclear force between nucleons depends on their spin orientation. It is too weak to bind two nucleons with anti-aligned spins and two identical nucleons (two protons) cannot be bound with identical spins because of the Pauli exclusion principle.
The nuclear force is a residual of the strong force between the quarks that make up nucleons. It is perhaps analogous to the van der Waals forces between atoms.
Fusion-type interactions cannot occur between electrons because there is no analogy to the nuclear force between electrons. Electrons (as far as we know) are point-like particles with no substructure, experiencing only the electromagnetic and weak forces.
The electromagnetic force between two electrons is clearly repulsive and cannot bind them. The weak force is much weaker than the electromagnetic interaction unless one could get the electrons to separations of $\leq 10^{-18}\ m$, which would require particle energies many orders of magnitude higher than in the centre of the Sun for example.
My (limited) understanding is that even then, the weak force cannot result in a bound state for isolated leptons (if anyone has a simple explanation that would be good here).
Electrons that are not isolated can form bound states - part of the basis for superconductivity - through long range electron-phonon interactions.
