Compositions, local atomic structures, porosity, and vibrational dynamics of carbons obtained by pyrolyzing epoxy novolak resins at 650 and 1000 degrees C are investigated using several neutron scattering techniques. The ultimate goal is to understand the origin of the very high capacity for Li uptake exhibited by these materials. Neutron radial distribution function analysis and incoherent inelastic scattering show that the structural motif is a planar hexagonal graphene fragment similar to 10 Angstrom on a side, with edge carbons terminated by single hydrogen atoms and random stacking between fragments. Small-angle neutron scattering reveals substantial porosity on a length scale comparable to the size of the graphene fragments. Coupled with computer simulations, these results are consistent with two proposed mechanisms for unusually high Li capacity one analogous to conventional intercalation but with Li on both sides of isolated graphene fragments, the other involving bonding of Li to H-terminated edge carbons.