The morphology and electronic properties of single and few-layer graphene films nanostructured by the impact of heavy high-energy ions have been studied. It is found that ion irradiation leads to the formation of nano-sized pores, or antidots, with sizes ranging from 20 to 60 nm, in the upper one or two layers. The sizes of the pores proved to be roughly independent of the energy of the ions, whereas the areal density of the pores increased with the ion dose. With increasing ion energy (>70 MeV), a profound reduction in the concentration of structural defects (by a factor of 2-5), relatively high mobility values of charge carriers (700-1200 cm2 V-1 s-1) and a transport band gap of about 50 meV were observed in the nanostructured films. The experimental data were rationalized through atomistic simulations of ion impact onto few-layer graphene structures with a thickness matching the experimental samples. We showed that even a single Xe atom with energy in the experimental range produces a considerable amount of damage in the graphene lattice, whereas high dose ion irradiation allows one to propose a high probability of consecutive impacts of several ions onto an area already amorphized by the previous ions, which increases the average radius of the pore to match the experimental results. We also found that the formation of "welded" sheets due to interlayer covalent bonds at the edges and, hence, defect-free antidot arrays is likely at high ion energies (above 70 MeV).