MoS2 nanooctahedra are believed to be the smallest stable closed-cage structures of MoS2, i.e., the genuine inorganic fullerenes. Here a combination of experiments and density functional tight binding calculations with molecular dynamics annealing are used to elucidate the structures and electronic properties of octahedral MoS2 fullerenes. Through the use of these calculations MoS2 octahedra were found to be stable beyond n(Mo) > 100 but with the loss of 12 sulfur atoms in the six corners. In contrast to bulk and nanotubular MoS2, which are semiconductors, the Fermi level of the nanooctahedra is situated within the band, thus making them metallic-like. A model is used for extending the calculations to much larger sizes. These model calculations show that, in agreement with experiment, the multiwall nanooctahedra are stable over a limited size range of 10(4)-10(5) atoms, whereupon they are converted into multiwall MoS2 nanoparticles with a quasi-spherical shape. On the experimental side, targets of MoS2 and MoSe2 were laser-ablated and analyzed mostly through transmission electron microscopy. This analysis shows that, in qualitative agreement with the theoretical analysis, multilayer nanooctahedra of MoS2 with 1000-25 000 atoms (Mo + S) are stable. Furthermore, this and previous work show that beyond similar to 10(5) atoms fullerene-like structures with quasi-spherical forms and 30-100 layers become stable. Laser-ablated WS2 samples yielded much less faceted and sometimes spherically symmetric nanocages.