Herein, we report on conductive ultrathin films (nanosheets) with the characteristics of stretchability and water vapor permeability for skin-conformable bioelectrodes. The films are fabricated by combining conductive fibrous networks of single-wall carbon nanotubes (SWCNTs) and poly(styrene-b-butadiene-b-styrene) (SBS) nanosheets (i.e., SWCNT-SBS nanosheets). An increase in the number of SWCNT coatings increases both the thicknesses and densities of the SWCNT bundles. The SBS nanosheets coated with three layers of SWCNTs (i.e., SWCNT 3rd-SBS nanosheets) show comparable sheet resistance to the SBS nanosheets coated with poly(3,4-ethylenedioxithiophene) doped with poly(4-styrenesulfonate acid) (PEDOT:PSS) containing 5 wt.% butylene glycol (i.e., PEDOT:PSS/BG5-SBS nanosheets). In addition, the SWCNT 3rd-SBS nanosheets exhibit significantly reduced elastic moduli and increased elongations at break compared to the PEDOT:PSS/BG5-SBS nanosheets. Furthermore, the calculated water vapor transmission ratio of the 210-nm-thick SBS nanosheets (268,172 g m−2 (2 h)−1) is greater than that of the filter paper (6345 g m−2 (2 h)−1). The SWCNT 3rd-SBS nanosheets attached to model skin show high tolerances to bending and artificial sweat at different pH values (i.e., the electrical resistance changes ~1.1 times). Finally, the SWCNT 3rd-SBS nanosheet is applied to detect the surface electromyogram from the forearm of a subject. This nanosheet displays a signal-to-noise ratio similar to that of the PEDOT:PSS/BG5-SBS nanosheet.