The fast blood stream of animals is associated with large shear stresses. Consequently, blood cells have evolved a special morphology and a specific internal architecture allowing them to maintain their integrity over several weeks. For instance, non-mammalian red blood cells, mammalian erythroblasts and platelets have a peripheral ring of microtubules, called the marginal band, that flattens the overall cell morphology by pushing on the cell cortex. In this article, we model how the shape of these cells stems from the balance between marginal band elasticity and cortical tension. We predict that the diameter of the cell scales with the total microtubule polymer, and verify the predicted law across a wide range of species. Our analysis also shows that the combination of the marginal band rigidity and cortical tension increases the ability of the cell to withstand forces without deformation. Finally, we model the marginal band coiling that occurs during the disc-to-sphere transition observed for instance at the onset of blood platelet activation. We show that when cortical tension increases faster than crosslinkers can unbind, the marginal band will coil, whereas if the tension increases slower, the marginal band may shorten as microtubules slide relative to each other.