The comment above responded to the OP by treating his/her question as trivial and followed it with a snarky "P.S. It's called ’statistics’." remark.
P.S. @qubex Some people don't have degrees and would actually like to work in data science regardless. The OP is asking about the practicality of that scenario.
We have statisticians (yes, plural) on my team who have published in Nature, and plenty with other backgrounds.
Even ignoring the data engineering side, there is plenty that statisticians don't do or know which is useful data science.
Take the two attitudes to p-tests, or what a "reasonable number of features" means. You drop the Jeff Dean "consider training models with billions of features" quote on a statistcians desk and see their eyes open.
Statistics is great, but data science is just as much programming as it is stats.
I wouldn't rule it out, but it would be useful to have accomplishments to point to, or a personal recommendation.
We have a stack of maybe 25 well qualified PhD data scientists waiting for interviews. It takes a lot to get to the head of that line.