I honestly fell into it by luck. I moved to NYC, studied machine learning in grad school, networked my ass off, and landed an internship.
From there I went full time as something of an ML engineer at a company with a strong tech culture, and learned as much as I could in both tech and ML/statistics. The rest is history (although I'm by no means a rockstar or whatever).
My path is hard to reproduce -- it starts with being in NYC or SF at a specific point in time, before the labor market became saturated with data science bootcamps and PhDs furiously learning Python while working on their dissertations.
Your best bet at this point is to produce a few data-related projects (maybe work on open source like scikit-learn and pandas?) and network like crazy. Someone somewhere will have a need for someone like you.