It is entirely possible to get into a Masters of Education program without having any sort of BA in education.
I am one example, my wife, another.
For me:
BA in History at State University of New York, College at Potsdam. I was then accepted into the Master's in the Science of Teaching program at the same school for Secondary Social Studies Ed (to be accepted into the MST programs, you only need to have a degree which addresses the content area. History was one of hte many that addressed Social Studies). Finish the program, take the state teaching tests, get certified to teach.
My wife got a B.A. in Art History. Tried the MST Childhood education thing, it didn't work out, so she changed to Educational Technology (an MSEd degree) for non-state certification.
So yeah, entirely possible.