Quantum confinement of positrons in nanoclusters offers the opportunity to obtain detailed information on the electronic structure of nanoclusters by application of positron annihilation spectroscopy techniques. In this work, positron confinement is investigated in lithium nanoclusters embedded in monocrystalline MgO. These nanoclusters were created by means of ion implantation and subsequent annealing. It was found from the results of Doppler broadening positron beam analysis that approximately 92% of the implanted positrons annihilate in lithium nanoclusters rather than in the embedding MgO, while the local fraction of lithium at the implantation depth is only 1.3 at. %. The results of two-dimensional angular correlation of annihilation radiation confirm the presence of crystalline bulk lithium. The confinement of positrons is ascribed to the difference in positron affinity between lithium and MgO. The nanocluster acts as a potential well for positrons, where the depth of the potential well is equal to the difference in the positron affinities of lithium and MgO. These affinities were calculated using the linear muffin-tin orbital atomic sphere approximation method. This yields a positronic potential step at the $\mathrm{MgO}\ensuremath{\Vert}\mathrm{Li}$ interface of 1.8 eV using the generalized gradient approximation and 2.8 eV using the insulator model.