The strong chemical bonds between C, N, and O play a central role in chemistry, and their formation and cleavage are critical steps in very many catalytic processes. The close-lying molecular orbital energies and large correlation effects pose a challenge to electronic structure calculations and have led to different bonding interpretations, most notably for C-2. One way to approach this problem is by strict benchmark comparison of related systems. This work reports reference electronic structures and computed bond dissociation enthalpies D-0 for C-2, CN, CN-, N-2, NO, NO+, O-2 and related systems C-2(+) and C-2(-), at chemical accuracy (similar to 1 kcal/mol or 4 kJ/mol) using CCSD(T)/aug-cc-pV5Z, with additional benchmarks of HF, MP2, CCSD, explicitly correlated F12 methods, and four density functionals. Very large correlation and basis set effects are responsible for up to 93% of total D-0. The order of the molecular orbitals 1 pi(u), and 3 sigma(g) changes, as seen in textbooks, depending on total and effective nuclear charge. Linear trends are observed in 2 sigma(u)-2 sigma(g) orbital splitting (R-2 = 0.91) and in D-0 of C-2, C-2(-), and C-2(+) (R-2 = 0.99). The correlation component of D-0 of C-2 is by far the largest (similar to 93%) due to a poor HF description. Importantly, density functional theory fails massively in describing this series consistently in both limits of effective nuclear charge, and Hartree-Fock exchange or meta functionals do not remedy this 100 kJ/mol error, which should thus be addressed in future density functional developments as it affects very many studies involving cleavage or formation of these bonds.