Nanoresonators in which mechanical vibrations and spin waves can be coupled are an intriguing concept that can be used in quantum information processing to transfer information between different states of excitation. Until now, the fabrication of freestanding magnetic nanostructures that host long-lived spin-wave excitations and may be suitable as mechanical resonators has seemed elusive. We demonstrate the fabrication of freestanding monocrystalline yttrium-iron-garnet (YIG) three-dimensional (3D) nanoresonators with nearly ideal magnetic properties. The freestanding 3D structures are obtained using a complex lithography process including room-temperature deposition and lift-off of amorphous YIG and subsequent crystallization by annealing. The crystallization nucleates from the substrate and propagates across the structure even around bends over distances of several micrometers to form, e.g., monocrystalline resonators, as shown by transmission electron microscopy. Spin-wave excitations in individual nanostructures are imaged by time-resolved scanning Kerr microscopy. The narrow line width of the magnetic excitations indicates a Gilbert damping constant of only alpha = 2.6 x 10(-4), rivaling the best values obtained for epitaxial YIG thin-film material. The fabrication process represents a step forward in magnonics and magnon mechanics as it provides 3D YIG structures of a high quality. At the same time, it demonstrates an alternative route toward the fabrication of freestanding crystalline nanostructures, which may be applicable also to other material systems.