I would suggest joining an early stage start up and getting involved with anything remotely to do with data science at every opportunity. I joined a small company as an analyst with no programming experience and minimal statistical knowledge. I was a graduate but not in a relevant subject and just taught myself the relevant skills on the side. It was a lot of work but not a waste of time. The programming side of the job can be learnt fairly quickly but the maths and stats side takes longer. I don't think you can really succeed in data science without both. Saying that, you certainly don't have to have a degree to be able to use that knowledge. I did just do a statistics degree though, and it has made the job a lot more pleasurable.