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The common picture of excitons in materials with atomic-like localization of electrons is that of Frenkel excitons, where electrons and holes stay close together, which is associated with a large binding energy. Here, using the example of the layered oxide V₂O₅, we highlight  another kind of exciton: it also has a huge binding energy but, at the same time, a large electron-hole distance. We explain that this seemingly contradictory finding is rooted in the charge transfer nature of the excitation. The anisotropy of the exciton delocalization is determined by the local anisotropy of the structure, whereas the exciton extends orthogonally to the chains formed by the crystal structure. Moreover we show that the bright exciton goes together with a dark exciton of even larger binding energy and more pronounced anisotropy. These findings are obtained by combining first principles many-body perturbation theory calculations, ellipsometry experiments, and tight binding modelling, leading to excellent agreement and a consistent picture. Our explanation is general and can be extended to other materials.