The roles of ribonucleoside and deoxyribonucleoside radicals in DNA and RNA damage cannot be properly understood in the absence of knowledge of the C-H and N-H bond dissociation enthalpies (BDEs) depicting the energy cost to generate each of these radicals. However, because the nucleoside radicals tend to be extremely short-lived and it is very difficult to separate and identify different nucleoside radicals, experimental BDEs for nucleosides have remained elusive. Herein, we developed an ONIOM-G3B3 method in order to reliably predict the BDEs of nucleosides and we carefully benchmarked this new method against over 60 experimental BDEs of diverse sizable molecules. It was found that the accuracy of the ONIOM-G3B3 method was about 1.4 kcal/mol for BDE calculations. Using the ONIOM-G3B3 method, a full scale of C-H and N-H BDEs were obtained for the first time for ribonucleosides and deoxyribonucleosides with an estimated error bar of +/- 1.4 kcal/mol. Discussions were then made about the interesting connections between these BDE values and previously reported experimental observations concerning radical-mediated DNA and RNA lesions. The significance of the work is twofold: (i) Nucleosides represent one of the most important groups of compounds in science. A full scale of reliable bond dissociation enthalpies for nucleosides is of fundamental importance. (ii) This work demonstrates the feasibility to accurately predict the bond strength of various sizable molecules ranging from nanosize molecular devices to biologically significant compounds.