Glass-forming ability in the (BaO)x(Al2O3)1−x system (0≤x≤1) was investigated by using the containerless aerodynamic levitation and laser-heating method. The main glass-forming region was found to occur for 0.40(2) ≤x≤ 0.48(2), where there is insufficient oxygen to form an ideal network of corner-sharing AlO4 tetrahedra in which the oxygen atoms are twofold coordinated, with another narrow glass-forming region at x = 0.62(2) around the eutectic composition. The glass corresponding to x = 0.4 was chosen for further investigation by using both neutron and x-ray diffraction, and a detailed atomistic model was built by applying a combination of molecular dynamics and reverse Monte Carlo methods. The results show a network structure based predominantly on corner-sharing tetrahedral AlO4 motifs in which triclusters (OAl3 units formed by three tetrahedral Al atoms sharing a common vertex) play an integral part, with as many as 21% of the oxygen atoms involved in these configurations. The barium ions bind to an average of 7.4 O atoms, most of which are twofold-coordinated bridging oxygen atoms. The larger size of barium compared to calcium narrows the range of glass-forming compositions in alkaline-earth aluminates such that the main glass-forming range corresponds to a regime in which an oxygen-deficient Al-O network is stabilized by the formation of triclusters.