Are real analysis and measure theory essential to learn for data science?
Neither of real analysis or measure theory are necessary for data science. Most of the mathematics you need to know is at the undergraduate level and these courses should suffice: 


Calculus 
Linear Algebra 
Probability and Statistics 
Optimization 


Most of machine learning involved in data science (that deals with application vs creating novel machine learning algorithms), requires a linear combination of concepts learnt in the above mathematics courses. For example, you can understand enough kernel methods well enough if you understood linear algebra (inner products, matrices etc.). 

Also, data science has to do with practice and experimentation. A lot of key concepts become clearer only through experimentation and if you need to understand a new method, the best way is to read up specifically about it and then experiment with it. In this respect, I feel studying an entire branch of involved mathematics is not worth it if your goal is to understand a machine learning method -- you can always ask a Guru in such situations to explain what you're missing at a conceptual level :-)