The structural designs of nonmagnetic chiral nihility-based optical fibers of elliptical and circular cross-sections, embedded with conducting sheath helix structure at the core–cladding interface, are investigated. Emphasis is given to the power confinement patterns while considering the hybrid EH01, EH11, and EH21 modes in the fiber and their comparative features. Variations in fiber dimensions are considered, and the corresponding effects on the power confinements under different orientations of sheath helix structure are reported. It has been found that the chiral nihility-based fiber structure results in the enhancement of the confined power, particularly in the case of an elliptic fiber. Apart from these, keeping in mind the possible lossy nature of chiral nihility mediums, the dispersive characteristic of material is also considered for both kinds of cross-sectional geometry, and the results are compared with situations when the mediums are lossless. The results essentially indicate less confinement of power in the case of lossy mediums.