Nah you don't need a Master's, many of the data scientist positions, especially at startups, don't require a MS or PhD (even though often its preferred).
Do lots of data science projects and of course, don't forget to meet people! Find local meetups and meet other data scientists. Write blog posts about a cool analysis on some data you did. Build libraries for Python, get involved in the community.
I would also focus way less on specific roles / titles/ 'becoming a data scientist'. Many companies have software engineers doing data science work. Others have positions like 'Data Engineers' where you split your time doing analysis and doing data infrastructure work. I would also consider Data Analyst and other positions as well.
The key is to find teams / companies that have smart people and are tackling interesting problems. Teams that promote curiosity and ones that have employees that are truly passionate about their work. In these kinds of environments, the titles are just there for HR. They are much more interdisciplinary and you can learn a lot from your coworkers on a variety of topics.
The company I'm interning for right now (as a data scientist) is looking for data scientists if you're interested. I haven't even finished my Bachelor's degree yet!
I've been interning since January and I've learned a lot more in 3 months than I would have in any academic Master's program. I've learned more interning as a data scientist than I did in my 4.5 years of undergraduate college!
