The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. This function can have a property called an infinite staircase, which implies infinitely many obstructions are relevant in determining whether embeddings exist. Based on various work with McDuff, Pires, and Weiler, we will discuss the classification of which Hirzebruch surfaces have infinite staircases. The argument relies on a correspondence between constructing embeddings via almost toric fibrations and finding obstructions via exceptional spheres. The talk will focus on explaining this correspondence.