We investigate the accuracy of first-principles many-body theories at the nanoscale by comparing the low-energy excitations of the carbon fullerenes C(20), C(24), C(50), C(60), C(70), and C(80) with experiment. Properties are calculated via the GW-Bethe-Salpeter equation and diffusion quantum Monte Carlo methods. We critically compare these theories and assess their accuracy against available photoabsorption and photoelectron spectroscopy data. The first ionization potentials are consistently well reproduced and are similar for all the fullerenes and methods studied. The electron affinities and first triplet excitation energies show substantial method and geometry dependence. These results establish the validity of many-body theories as viable alternative to density-functional theory in describing electronic properties of confined carbon nanostructures. We find a correlation between energy gap and stability of fullerenes. We also find that the electron affinity of fullerenes is very high and size independent, which explains their tendency to form compounds with electron-donor cations.