α-graphdiyne is a novel predicted Dirac cone material, which is similar to graphene. But the absence of a band gap significantly limits its practical applications. In order to extend this limitation, an opening of energy gap is needed. To this end, we resort to the nanoribbon structure of α-graphdiyne. This is a conventional proposal to open up the energy gaps in nanomaterials. The results show that both the armchair and the zigzag α-graphdiyne nanoribbons do generate energy gaps, which are width-dependent. In addition, the underlying mechanism of this opening is explored. The former is ascribed to the combination of quantum confinement and edges' effect, while the latter arises from the edge magnetic ordering. These novel nanoribbons with opening energy gaps would be potentially used in electronic devices.