We report the observation of Peierls-type metal-insulator transition in carbon nanostructures formed by chemical vapor deposition inside the pore network of the ZSM-5 zeolite. The Raman spectrum of this nanocarbon@ZSM-5 indicates a clear signature of the radial breathing mode (RBM) for (3,0) carbon nanotubes that can constitute the carbon network segments. Electrical transport measurements on multiple few-micron-sized nanocarbon@ZSM-5 crystals showed metallic temperature dependence of resistance down to 30 K, at which point the resistance exhibited a sharp upturn that is accompanied by the opening of a quasigap at the Fermi level as indicated by the differential resistance measurements. Further Hall measurements have yielded both the sign of the charge carrier and its density. The latter demonstrated excellent consistency with the quasigap data. We employed first-principles calculations to verify that there can indeed be softening of the phonon modes in the (3,0) carbon nanotubes.