We point out that the Coulomb part of the QED Hamiltonian in the Coulomb gauge has exact two-fermion eigenstates, provided that the wavefunction satisfies a Dirac-like (or Breit-like) equation. This equation, which describes the relative motion of a system of two fermions of masses and and charges and interacting via the Coulomb potential, is shown to reduce to the usual Dirac eigenvalue equation when one of is taken to be infinite. For specific states of the two-fermion systems, the equation is reduced to coupled radial equations. Numerical solutions for the mass spectrum of the two-fermion system as a function of the coupling constant are obtained for states for various combinations of and . We find that the ground-state energy of the two-fermion system has normalizable bound-state solutions for , where for , but decreases towards the one-particle Dirac result of as one of the particle masses tends to infinity. Our numerical results for are in agreement with conventional perturbative results if . Comparison is made with other radial reductions of two-fermion equations with purely Coulombic interactions.