The ability to define an off state in logic electronics is the key ingredient that is impossible to fulfill using a conventional pristine graphene layer, due to the absence of an electronic bandgap. For years, this property has been the missing element for incorporating graphene into next-generation field effect transistors. In this work, we grow high-quality armchair graphene nanoribbons on the sidewalls of 6H-SiC mesa structures. Angle-resolved photoelectron spectroscopy (ARPES) and scanning tunneling spectroscopy measurements reveal the development of a width-dependent semiconducting gap driven by quantum confinement effects. Furthermore, ARPES demonstrates an ideal one-dimensional electronic behavior that is realized in a graphene-based environment, consisting of well-resolved subbands, dispersing and non-dispersing along and across the ribbons respectively. Our experimental findings, coupled with theoretical tight-binding calculations, set the grounds for a deeper exploration of quantum confinement phenomena and may open intriguing avenues for new low-power electronics.