The present work aims at development of self-healing materials capable of partially restoring their mechanical properties under the conditions of prolonged periodic loading and unloading, which is characteristic, for example, of aerospace applications. Composite materials used in these and many other applications frequently reveal multiple defects stemming from their original inhomogeneity, which facilitates microcracking and delamination at ply interfaces. Self-healing nanofiber mats may effectively prevent such damage without compromising material integrity. Two types of core-shell nanofibers were simultaneously electrospun onto the same substrate in order to form a mutually entangled mat. The first type of core-shell fibers consisted of resin monomer (dimethylsiloxane) within the core and polyacrylonitrile within the shell. The second type of core-shell nanofibers consisted of cure (dimethyl-methyl hydrogen-siloxane) within the core and polyacrylonitrile within the shell. These mutually entangled nanofiber mats were used for tensile testing, and they were also encased in polydimethylsiloxane to form composites that were also subsequently subjected to tensile testing. During tensile tests, the nanofibers can be damaged in stretching up to the plastic regime of deformation. Then, the resin monomer and cure was released from the cores and the polydimethylsiloxane resin was polymerized, which might be expected to result in the self-healing properties of these materials. To reveal and evaluate the self-healing properties of the polyacrylonitrile-resin-cure nanofiber mats and their composites, the results were compared to the tensile test results of the monolithic polyacrylonitrile nanofiber mats or composites formed by encasing polyacrylonitrile nanofibers in a polydimethylsiloxane matrix. The latter do not possess self-healing properties, and indeed, do not recover their mechanical characteristics, in contrast to the polyacrylonitrile-resin-cure nanofiber mats and the composites reinforced by such mats. This is the first work, to the best of our knowledge, where self-healing nanofibers and composites based on them were developed, tested, and revealed restoration of mechanical properties (stiffness) in a 24 h rest period at room temperature.