Exact sequences occur in many parts of mathematics. The shorter ones can arise in many different ways. However, if you have an exact sequence of length six or more, it can usually be interpreted naturally using homological algebra, as the long exact sequence associated to a short exact sequence of chain complexes. There are some sequences that cannot be interpreted that way, but essentially all of those can be understood as the long exact sequence of homotopy groups associated to a suitable fibration. Typically, there is no other useful way to understand them. Moreover, the Dold-Kan theorem allows us to convert any short exact sequence of chain complexes to a fibration of spaces. Thus, the theory of fibrations and homotopy groups is essentially the simplest context in which we see all naturally occurring long exact sequences.