We have studied hydrogen-passivated amorphous carbon nanostructures with semiempirical molecular orbital theory in order to provide an understanding of the factors that affect their electronic properties. Amorphous structures were first constructed using periodic calculations in a melt/quench protocol. Pure periodic amorphous carbon structures and their counterparts doped with nitrogen and/or oxygen feature large electronic band gaps. Surprisingly, descriptors such as the elemental composition and the number of sp3-atoms only influence the electronic structure weakly. Instead, the exact topology of the sp2-network in terms of effective conjugation defines the band gap. Amorphous carbon nanodots of different structures and sizes were cut out of the periodic structures. Our calculations predict the occurrence of localized electronic surface states, which give rise to interesting effects such as amphoteric reactivity and predicted optical band gaps in the near-UV/visible range. Optical and electronic gaps display a dependence on particle size similar to that of inorganic colloidal quantum dots.