Q:

Why do electron-positron pair annihilate upon contact?

I'll a appreciate a layman's explanation, if there exists one, to this question that arose when reading an popular-science level article on Einstein and the $E=MC^2$ equation.
What I mean is that, why is the product of the reaction not an electrically neutral doublet of mass $2M_e$ (= 1.022 MeV) and which the opposite charges keep the two particles attracted to each-other?

A:

An explanation for a layman.
A hydrogen atom exists because of quantum mechanics, and not classical mechanics. In quantum mechanics there exist energy levels where the electron of the hydrogen atom, which has a tiny mass with respect to the proton, can only occupy  fixed energy levels around the proton. The lowest one is the ground state and is non zero, allowing for the existence of the hydrogen atom. Otherwise, if we only had the classical theory, the electron would fall onto the proton and no hydrogen atom would be possible. In the quantum ground state,  the electron  does not have enough energy to change a proton into a neutron, energy conservation prevents the disappearance.
Still, in quantum mechanics, which is a probabilistic theory, there may exist a tiny probability for the electron wave function to meet a quark within the proton nucleus and interact, with the effect of destroying the hydrogen atom. The probability is so small that it can be ignored.
In contrast, the positronium "atom" has energy levels like hydrogen , and that is why we observe it,  but the probabilities for the electron and the positron at ground state to overlap is huge, because they have equal masses and interact mainly with the electromagnetic field, and once the positron reaches the ground state (corresponding to the ground state of the hydrogen electron mathematically) it overlaps with the electron and annihilates because that is the most probable outcome. 
It is all a matter of  sizes of constants describing the interactions, and conservation laws, whose details you need to study to comprehend.

