Yes, you can make genealogy your career. Becoming a professional genealogist takes a lot of time and effort. Here are some resources to get you started.
What is the Genealogical Proof Standard?
The Genealogical Proof Standard (GPS) is a guideline for establishing the reliability ("proof") of a genealogical conclusion with reasonable certainty. It is important within the genealogical community for clearly communicating the quality of research performed, such as by a professional genealogist.
Becoming certificated or obtaining a degree in genealogy will provide you with the skills not only to research your own family history, but to pursue it as a career as well.