In graphene nanoribbons (GNRs), the lateral confinement of charge carriers opens a band gap, the key feature to enable novel graphene-based electronics. Successful synthesis of GNRs has triggered efforts to realize field-effect transistors (FETs) based on single ribbons. Despite great progress, reliable and reproducible fabrication of single-ribbon FETs is still a challenge that impedes applications and the understanding of the charge transport. Here, we present reproducible fabrication of armchair GNR-FETs based on a network of nanoribbons and analyze the charge transport mechanism using nine-atom wide and, in particular, five-atom-wide GNRs with unprecedented conductivity. We show formation of reliable Ohmic contacts and a yield of functional FETs close to unity by lamination of GNRs on the electrodes. Modeling the charge carrier transport in the networks reveals that this process is governed by inter-ribbon hopping mediated by nuclear tunneling, with a hopping length comparable to the physical length of the GNRs. Furthermore, we demonstrate that nuclear tunneling is a general charge transport characteristic of the GNR networks by using two different GNRs. Overcoming the challenge of low-yield single-ribbon transistors by the networks and identifying the corresponding charge transport mechanism puts GNR-based electronics in a new perspective.