We consider Dirac operators with a Coulomb-type potential $V(x)\sim -\nu/|x|$, in the case where the Coulomb singularity is strong, that is $\sqrt3/2\leq\nu\leq1$ in units such that $mc^2=1$. This operator is not essentially self-adjoint but has a distinguished self-adjoint extension. In a first part we obtain new results on the domain of the distinguished self-adjoint extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the gap, in simple function spaces that are independent of the value of $0\leq\nu\leq1$. Our results include the critical case $\nu=1$ and they are the first in this setting. We also give the corresponding results in two dimensions.