Carbon nanomaterials, such as carbon nanotubes (CNTs) and carbon nanofibers (CNFs), are chemically inert in their highly graphitic forms. Various post processing methods can activate their surfaces to enhance their interactions with a host matrix in a nanocomposite. Chemical surface functionalization is used often. This method however can lead to major strength loss in nanomaterials stemming from induced surface defects (changing sp2 bonds to sp3 bonds). In this manuscript, we have experimentally studied the mechanical properties of the individual, pyrolysis-fabricated CNFs. These CNFs have a highly crosslinked 3D network of C-C bonds. The strength of CNFs has been studied as a function of O/C ratio. The loss in strength due to functionalization has been compared to that of other carbon nanomaterials with layered strcutures (CNT and graphene). Comparisons were also made with carbon microfibers. Fracture strength estimations of the critical flaw size in CNFs, CNTs and graphene were also made. The results revealed that despite having high surface area, carbon nanomaterials with crosslinked microstructure are resilient to flaws as big (deep) as 10-30 nm, while nanomaterials with layered structure (such as CNTs) experience a dramatic loss in strength with much lower flaw sizes. Hence, it seems that graphitic nanomaterials such as graphene and CNT have high strenght that, although higher than CNFs, comes at a cost to flaw tolerance and robustness. Since failure is often progressive, this work demonstrates a benefit that crosslinked nanomaterials have over highly graphitic ones, such as CNTs, in load bearing applications.