As a museum professional myself, I'd tell you that it depends entirely on the type of museum you're looking to work for, and the sort of position you're interested in. My answers are based on my experience in the USA - YMMV in other countries with different museum funding and staffing models. My answers are also biased towards larger museums, though I try to comment on smaller ones where I feel that I know enough to do so.

Curators - the subject matter experts in their various departments and fields - are responsible for original research and collection connoisseurship. They typically will have a Ph.D specific to their field if such a thing is available, or a doctorate in a field with a related personal focus if not. This is generally the case regardless of the size of the museum.

Collections staff - the folks responsible for the daily care of the material objects and day to day functioning of the various internal divisions - will usually either have a Masters in Museum Studies, Library Science, or (in large museums) in a scientific or historical field related to the specific department they are in. Those with the Museum or Library graduate degrees will typically have undergraduate degrees in a field broadly related to something of interest at their museum (History, Art History, hard sciences, etc)

Registrars - those in charge of tracking the massive amounts of paperwork, legal and otherwise, associated with possession and movement of collections - are likely to have graduate degrees in either Museum, Library, or Archival fields, and/or have a legal background, regardless of the size of a museum.

Conservators - those responsible for the physical maintenance and restoration of collections - have many years of training specific to their sub-specialty of collections, either in a graduate program or apprenticeship, on top of an undergraduate background that typically includes chemistry, physics, and art. They tend to only be on staff at large museums, with smaller ones hiring private practitioners as needed.

Archivists - those responsible for retaining the long term non-legal records of institutions, as well as collecting and maintaining collections of documentary records (such as papers, recordings, and video media) - will typically have graduate degrees in Archival or Library Science, or else Museum Studies with significant classwork and experience in an archival environment. Archives are often special entities separate from museums, though can also be internal branches of larger museum institutions. In some specialty archives that focus on very specific topics, graduate degrees in those topics may be desirable.

The job market in the world of museums is incredibly tight at present, at least in the USA. Simply having a degree is meaningless without the experience or connections from internships and volunteering to get your foot in the door. I know literally nobody in my entire professional circle who didn't do this first. If you've got an MA in Medieval Studies (and a BA in a something similar), the primary path I think would be open in larger institutions would be through either an art museum or archive that includes specialization in your field, and which already has a lead archivist or lead curator in the division who is in need of assistance and has the funding to get it. There, your subject matter knowledge may be enough to get you the chance you need as an assistant curator or assistant archivist, on the assumption that you can learn the more specific techniques of museums or archives on the job. At a small museum, especially the local history types that are often operated by a single dedicated staffer with the assistance of volunteers or part-timers when available, you may be in luck as well - your academic credentials and ability to research and organize your thoughts in cogent ways to present historical information could be just what they're looking for.

The best way you can increase your chances, regardless of size, is to physically get in to the museums as a volunteer or intern, to learn the ropes, meet people, and get an inside track on any potential job postings. There is absolutely no substitute for hands-on experience and getting your face known. Expect that for the first while, you might be a private contractor, stringing together jobs from one small contract to another for years or more, building your reputation and waiting for an appropriate opening to occur at a place you want to work. It isn't a glamorous life, but it can be rewarding if you're certain it's what you want to do long term. For most museum professionals, their belief in what they do and the thrill of what they work with is what makes them accept wages and benefits that can be far below what one otherwise might expect for the amount of education and training they have.

The academic credential is probably required. But looking at my friends who have earned various medievalish degrees in the last few years: the decisive factor in obtaining an archival job is hands-on manuscript experience. So, whatever program gives you the paleography and codicology knowledge, skills, and opportunity to demonstrate them in your thesis research. A summer internship in the manuscript collection at the British Library or stateside equivalents also seems to help.

I should add that not a lot of people I know have had success with museum jobs, at least not the interfacing with the public type. It's skewed much more towards library, archive, and even auctionhouse assessment work.

The type of medieval degree isn't as important, in my experience, as also having experience in museum or library work. And quite possibly an MLIS degree or a MA in Public History.