Nice! Yeah, I think it's really important
not to be intimidated by stats & data science. I mean, as ihnorton said, if you want a career in academia, you need an academic degree. Period. You want to enter a v sophisticated field, like quantitative finance or Google search or whatever, that degree is really going to help. But if you just want to surround yourself with good data, good questions, data-loving friends, then I believe you can teach yourself. Absolutely.
Why do I think this? Because (1) the day-to-day of "data science" is more of a craft than a science. Before you run complicated analyses, you are just cleaning the data, visualizing it, trying to see what questions it could answer. It feels a lot like playing in the woods to me... you're just knocking about, seeing what cool stuff is in there, building a fort out of scatterplots. And a truly wise data analyst will know when to not run tests at all. Also (2) it would surprise you how many academics are themselves kinda self-taught. They get into research because of their interest in a specific topic, and only come around to learning stats later. Think about it! The MCAT does not test prob/stats, even though doctors end up reading reams of statistical studies for work. I'm not saying these doctors are ignorant, not at all... just that after undergrad, they (and a bunch of PhD candidates) find themselves in the exact same position you're in. They teach themselves, or they squeeze into stat dept courses, and they all turn out OK.
I would just look for the data that's already around you. Cool data being produced at work? Ask if you can make some charts in R/Python/Julia to show your team. The internship suggestion was a great idea. Or you can take on some cool longish-term project and blog your way through it online. You'll do the next thing, then the next thing, and so long as you keep the awe, you'll end up somewhere neat!