The electrostatic repulsion between me and the chair keeps me hovering above the chair at all times.
What is the average distance that exists between any two everyday object due to electrostatic repulsion and gravity? How can this distance be determined?
There is some arbitrary definition of "hovering," or rather the reference position where you would no longer be hovering but touching involved: Since atoms' (and molecules') electron wavefunctions extent, although exponentially decaying, to infinity, you are in a sense touching and even partially overlapping every atom at the edge of the visible universe! The question is how much you overlap or how close the atomic nuclei get to result in sufficient repulsion to counteract your weight.
If you are happy with a very rough, barely order-of-magnitude calculation, you could use the difference in energy and typical electron-nucleus distances between atomic orbitals as an approximation to the spring constant (force per distance) involved. Note that this is really, really rough, essentially assuming a quadratic pseudopotential that would not result in any atomic orbitals at all, but as a first approximation, I'll use it anyways. For light atoms (say hydrogen), that will give you something on the order of $10\,\textrm{eV}/(0.1\,\textrm{nm})^2$ or (rounded to zero significant digits) 100 pN/pm per atom.
Assuming that you have a surface full of atoms that each cover $(0.1\,\textrm{nm})^2$ of a total contact area of a well-padded butt of a square foot, each of them needs to be deflected about one attometer, which is $10^{-18}$ meters or one billionth of a nanometer), to produce the order of magnitude of a person's weight, $10^3\,\mathrm{N}$. If you take your zero to be the zero-force distance (for example after adhesion of fat or water molecules but before applying your weight), then this is not your hovering height but the depression (in the opposite direction) that results.
Note that the real situation may differ significantly: If, for example, only one in a thousand atoms on the surfaces "touch" sufficiently closely to contribute to this force, the rest must bear a force a thousand times higher and hence, in this crude model, get deflected a thousand times more. The only justification for using this model (Hooke's law or a quadratic potential) is that almost any real effective potential will, expanded around the rest point where the first derivative vanishes, have a second derivative (quadratic shape) as first non-vanishing approximation. By going rather far with this approximation in using excited states to get an order-of-magnitude estimation, already a significant (more than an order of magnitude?) error is introduced, and this is only considering Pauli exclusion effects (and electrostatics between electrons and their own nucleus), not the electrostatics between different atoms that you specifically mentioned.
