Yes, definitely! No matter what degree you have, it's unlikely. It's one of the careers with the highest oversupply since so many individuals want to work in it yet there are so few actual openings. It is referred to as the "CSI effect."

However, Biomed and Biochem would both give you a wide range of employment alternatives, especially Biochem, which is the harder and, from what I can tell, less enjoyable degree (there are no opportunities that a Biomed can acquire that a Biochem cannot, but the opposite is not true). A more focused field would be medicinal chemistry.

Hope I have answered your question. All the best!