For a data science role focused around exploratory analysis, model building and idea generation (as opposed to a more engineering-focused data scientist) I'd say that a basic understanding of programming concepts and familiarity with the most essential tools (version control, Linux, Python/R) is probably enough to be accepted into a data science role. Data science degree programs are still quite novel so most companies don't expect to recruit people with such a degree, hence there are good opportunities for people with a STEM education I'd say.
Indeed, plus you have zero business experience straight out of such a program so you are at a disadvantage if you are starting your career as such.
I only have my experience and don't know how repeatable it is, but I just got here by building stuff that I thought was interesting.
Right at the end of college I tried to build a connected hardware/data harvesting startup, and it failed because I was a naive idiot who didn't understand just how hard the financials of building a hardware company were (all of the money up front for manufacturing runs) and how much time I would have to spend on things that I wasn't good at or really interested in, like building out supply chain and design for manufacturing or dealing with radio emission problems.
But over that time I collected a couple of relevant awards (decently sized pitch contests, won at a mlh hackathon I did for fun etc.), built interesting prototypes of the hardware that worked and a web platform that supported it that people thought was interesting and that all worked as a hedge to show that I can at least build interesting things. So when I looked a recruiter came to me with a software engineering job at a startup that I thought was novel and interesting and I ended up taking it.
Once I was there I saw a problem with a lot of labor going into dealing with relations between a bunch of types of texts, so I built a preliminary thing to automate as much of that as possible using relatively basic clustering and putting a person in the loop to collect labels so that I can come back and try something supervised later. The CEO brought on a chief data scientist to help me fix everything that I messed up and build a more scalable deployment pipeline and architecture for datascience microservices and now I work with him.
But yeah, my experience is just that I built things and tried to make sure they worked and were sane, and then just showed people those things to convince them to let me do that for other things I wanted to do. I have a couple friends who did relatively similar things with pretty similar results.