Fiction work does not require a degree to get published. But of course, having a degree in anything in your resume gives you more credit to both the agent and the publisher's eye. Also, though story matters, it is the marketability of the story that counts - a.k.a how much will it sell.
Non-fiction works obviously require a degree or a credential of sorts. A non-doctor will not be taken seriously if he or she is trying to publish a medical literature.
Anyway, getting published is one of the hardest thing right now. Edit your work a couple of times and avoid mistakes. Then, get an agent and let them find an interested publisher for you.