In many developmental and pathological processes, including cellular migration during normal development and invasion in cancer metastasis, cells are required to withstand severe deformations. The structural integrity of eukaryotic cells under small deformations has been known to depend on the cytoskeleton including actin filaments (F-actin), microtubules (MT) and intermediate filaments (IFs). However, it remains unclear how cells resist severe deformations since both F-actin and microtubules fluidize or disassemble under moderate strains. Using vimentin containing IFs (VIFs) as a model for studying the large family of IF proteins, we demonstrate that they dominate cytoplasmic mechanics and maintain cell viability at large deformations. Our results show that cytoskeletal VIFs form a stretchable, hyperelastic network in living cells. This network works synergistically with other cytoplasmic components, substantially enhancing the strength, stretchability, resilience and toughness of cells. Moreover, we find the hyperelastic VIF network, together with other quickly recoverable cytoskeletal components, form a mechanically robust structure with a self-healing nature.