The formally real Jordan algebras include the class of JB-algebras, a class of normed Jordan algebra which is the Jordan algebra equivalent of C*-algebras. From this you might imagine that obtaining a complete classification is a rather non-trivial task. In this class you also find JW-algebras, which are JB-algebras with a predual, thus corresponding to von Neumann algebras. Some of the concepts from C*- and von Neumann algebra theory carry over to the Jordan algebra setting, but this margin is too narrow to summarize what is known. With apologies for tooting my own horn here, in 1984 I coauthored a book “Jordan Operator Algebras” with Erling Størmer which pretty much summarized the state of the art at the time. (Since then I have left that field, so I don't know if a lot has happened since.)