Computing the diameter, and more generally, all eccentricities of an undirected graph is an important problem in algorithmic graph theory and the challenge is to identify graph classes for which their computation can be achieved in subquadratic time. Using a new recursive scheme based on the structural properties of median graphs, we provide a quasilinear-time algorithm to determine all eccentricities for this well-known family of graphs. Our recursive technique manages specifically balanced and unbalanced parts of the Θ-class decomposition of median graphs. The exact running time of our algorithm is O(n log^4 n). This outcome not only answers a question asked by Bénéteau et al. (2020) but also greatly improves a recent result which presents a combinatorial algorithm running in time O(n^1.6408 log^{O(1)} n) for the same problem. Furthermore we also propose a distance oracle for median graphs with both polylogarithmic size and query time. Speaking formally, we provide a combinatorial algorithm which computes for any median graph G, in quasilinear time O(n log^4 n), vertex-labels of size O(log^3 n) such that any distance of G can be retrieved in time O(log^4 n) thanks to these labels.