How is the potential energy between two atoms measured?
The morse potential (equation see e.g. wikipedia) basically contains the bond dissociation energy, a "force constant" and the bond length at ground state. It does not give any measurable reality, but is just a mathematical model describing (approximating) the same.

The lower part of the curve (usually the ground state and perhaps the first two vibrationally excited states) are approximated by the harmonic oscillator, the energies of which you can get from  IR or Raman spectroscopy. The harmonic oscillator (= an ideal spring) has only one parameter, the force constant.

The bond lengths you can get either directly from microwave rotational spectroscopy, or from the rotational fine structure of you IR spectrum. They give you the reduced mass of the structure, which gives you the bond length if you know the absolute mass of the two atoms.

For the math and actual application, I refer you to any standard physical chemistry textbook.