Multimodal entity linking (MEL) is the task that consists of matching textual and visual mentions of entities in unstructured data to their corresponding entities in a knowledge base (KB). To be effective in large-scale practical settings, MEL systems must meet three objectives: high linking accuracy, computational efficiency, and storage efficiency, i.e., a compact yet efficient index of the KB. In this paper, we highlight that state-of-the-art systems fail to simultaneously satisfy these 3 requirements. To meet this three-fold objective, we propose FAST-MEL, a lightweight encoder-based MEL solution that relies on a novel and compact fixed-size vectorized representation of both the textual and visual information of each entity or mention. It matches the accuracy of the best systems but performs three orders of magnitude faster. It also consumes one order of magnitude less storage than the fastest systems.