We use numerical simulations to study the phase behavior of a system of purely repulsive soft dumbbells as a function of size ratio of the two components and their relative degree of deformability. We find a plethora of different phases, which includes most of the mesophases observed in self-assembly of block copolymers but also crystalline structures formed by asymmetric, hard binary mixtures. Our results detail the phenomenological behavior of these systems when softness is introduced in terms of two different classes of interparticle interactions: (a) the elastic Hertz potential, which has a finite energy cost for complete overlap of any two components, and (b) a generic power-law repulsion with tunable exponent. We discuss how simple geometric arguments can be used to account for the large structural variety observed in these systems and detail the similarities and differences in the phase behavior for the two classes of potentials under consideration.