Perfectly aligned, long dipolar hard rods are shown to exhibit unusual phase behavior, induced by the long range of the dipolar interaction in combination with the finite length of the particles. Three variants have been considered: (i) ellipsoids with central dipoles, (ii) spherocylinders with central dipoles, and (iii) spherocylinders with dipoles placed at regular intervals along their axes. In all cases, the dipoles are taken to be pointlike and directed longitudinally. At sufficiently low temperatures, coexistence between fairly low-density phases of similar structure has been found, which terminates at a critical point. Our results shed some light on recent simulations of dipolar soft spheres in a strong field, which separate into two "gas" phases of ordered, rodlike chains: Accord is semiquantitative in case (iii) and qualitative in cases (i) and (ii). Relaxing the assumptions of perfect order and molecular rigidity worsens agreement somewhat, but otherwise leads to no substantial changes. Possible refinements of the theory are discussed.