  Models of Asymmetric Dark Matter (ADM) with a sufficiently attractive and
long-range force gives rise to stable bound objects, analogous to nuclei in the
Standard Model, called nuggets. We study the properties of these nuggets and
compute their profiles and binding energies. Our approach, applicable to both
elementary and composite fermionic ADM, utilizes relativistic mean field
theory, and allows a more systematic computation of nugget properties, over a
wider range of sizes and force mediator masses, compared to previous
literature. We identify three separate regimes of nugget property behavior
corresponding to (1) non-relativistic and (2) relativistic constituents in a
Coulomb-like limit, and (3) saturation in an anti-Coulomb limit when the
nuggets are large compared to the force range. We provide analytical
descriptions for nuggets in each regime. Through numerical calculations, we are
able to confirm our analytic descriptions and also obtain smooth transitions
for the nugget profiles between all three regimes. We also find that over a
wide range of parameter space, the binding energy in the saturation limit is an
${\cal O}(1)$ fraction of the constituent's mass, significantly larger than
expectations in the non-relativistic case. In a companion paper, we apply our
results to synthesis of ADM nuggets in the early Universe.
