Recent progress in nanoscale optical physics is associated with the development of a new branch of nanophotonics exploring strong Mie resonances in dielectric nanoparticles with a high refractive index. The high-index resonant dielectric nanostructures form building blocks for novel photonic metadevices with low losses and advanced functionalities. However, unlike extensively studied cavities in photonic crystals, such dielectric resonators demonstrate low quality factors (Q factors). Here, we uncover a novel mechanism for achieving giant Q factors of subwavelength nanoscale resonators by realizing the regime of bound states in the continuum. In contrast to the previously suggested multilayer structures with zero permittivity, we reveal strong mode coupling and Fano resonances in homogeneous high-index dielectric finite-length nanorods resulting in high-Q factors at the nanoscale. Thus, high-index dielectric resonators represent the simplest example of nanophotonic supercavities, expanding substantially the range of applications of all-dielectric resonant nanophotonics and meta-optics.