It depends on how you define "data science".
If you are like AWS and say that using logistic regression is machine learning, then yes, you can teach yourself data science. Learn SQL, read a couple of books on logistic regression, use some open data for building a couple of models. There are many companies where you can have a decent job and an easy living with SQL and logistic regression on your tool belt.
If you say that data science starts with automating stock trading or building the intelligence of self driving cars, than no, you can not teach yourself data science. You will need at least one degree. Or more.