Graphene nanoribbons will be essential components in future graphene nanoelectronics. However, in typical nanoribbons produced from lithographically patterned exfoliated graphene, the charge carriers travel only about ten nanometres between scattering events, resulting in minimum sheet resistances of about one kilohm per square. Here we show that 40-nanometre-wide graphene nanoribbons epitaxially grown on silicon carbide are single-channel room-temperature ballistic conductors on a length scale greater than ten micrometres, which is similar to the performance of metallic carbon nanotubes. This is equivalent to sheet resistances below 1 ohm per square, surpassing theoretical predictions for perfect graphene by at least an order of magnitude. In neutral graphene ribbons, we show that transport is dominated by two modes. One is ballistic and temperature independent; the other is thermally activated. Transport is protected from back-scattering, possibly reflecting ground-state properties of neutral graphene. At room temperature, the resistance of both modes is found to increase abruptly at a particular length--the ballistic mode at 16 micrometres and the other at 160 nanometres. Our epitaxial graphene nanoribbons will be important not only in fundamental science, but also--because they can be readily produced in thousands--in advanced nanoelectronics, which can make use of their room-temperature ballistic transport properties.