We investigate the structural properties of Ge nanostructures selectively grown on Si. Defect-free nanostructures with a lateral size of 100 nm and surrounded by a thick (similar to 20 times larger than the coherent-film limit) Ge layer are achieved as demonstrated by transmission electron microscopy. As demonstrated by modeling based on elasticity theory solved by finite element methods, the peculiar combination of morphology and chemical composition of the nanostructures allows for a very efficient elastic relaxation of the heteroepitaxial strain. We demonstrate that, despite the relatively large size of the nanostructures, even a single dislocation would raise the energy of the system. A direct comparison between the strain field predicted by modeling and measured by energy-dispersive synchrotron-radiation grazing incidence x-ray diffraction shows substantial agreement.