Multicompartmental “core–sheath” fibers composed of a poly­(caprolactone) (PCL) polymer sheath and poly­(ethylene glycol) (PEG) fluids as the core materials were designed via coaxial electrospinning. Mechanical stretching of the fibers caused a discontinuous mechanical damping or stiffening behavior when the cores were composed of a PEG fluid as a known non-Newtonian shear thickening fluid (PEG and SiO2 particles). Surprisingly, it is found that shear thickening fluids are not a requirement for mechanical damping as is evidenced by similar behavior with Newtonian viscous PEG liquids. Data from optical microscopy, thermogravimetric analysis, dynamic mechanical analysis, and rheology have been employed to gain insights into the interactions between the PCL sheath and the PEG cores. The degree of mechanical damping was found to correlate with the viscosity of the core PEGs and is discussed in terms of the interactions between the core and sheath during mechanical oscillation. In addition, the nonwoven fiber mats were tested for auditory sound attenuation (e.g., white noise, pink noise, frequency steps, and chirps). The fiber mats effectively attenuate sound, especially in the low-frequency regions where their ability to dissipate energy is most prevalent. It is also clear that the degree of sound attenuation is dependent upon the core liquid viscosity. To the best of our knowledge, the results presented here are the first report of mechanical damping behavior in electrospun core–sheath fibers that employ liquid cores to attenuate auditory sound.