There is a box on the initial application that needs to be checked off (it's something along the lines of "Are you Jewish"), but you can manage to get on the program without having any close Jewish relatives. I think it's worth mentioning though that the majority of the people who go on MASA programs will be Jewish with some religious background (Reconstructionist, Reform, or Conservative), and...
TALMA is a more TEFL-oriented program in Israel that goes for about one month. I would highly recommend checking it out! http://www.talmaisrael.com/