This is going to depend on three factors.

Do you plan on doing PhD research in Early Medieval History? If the answer to this is yes, then you have your work cut out for you. Most PhD programs are going to expect you come into the program with a working knowledge of Latin at the very least. Ideally some familiarity with a modern European language such as German or French as well. And many, most if not all really, programs will make you take language exams to prove your familiarity and expertise with the languages. Two or three languages in addition to English are not simple expected, but required.

Or is this an effort to be more familiar with the state of the field as a whole? In this case, I would advise to focus on learning one, at the most two, additional languages such as Latin or French, and after you've reached a level of familiarity with one, then you can add another. One modern language and one Medieval one would be a lot of work as a hobby. However to be truthful, if this is a casual interest, then studying an entire language or two seems a bit excessive. If you are looking to do Master's level work, then a language or two will probably suffice.

And finally, what area of the Medieval that you want to focus on? This really does decide what languages you are going to be working with. For example if you want to focus in on Norse studies, then Old Norse and a modern Scandinavian language such as Swedish or Norwegian would be advisable. If you wanted to study the Middle East, then Arabic would be necessary, and so on and so forth.