[shas3]

Yitang Zhang is an outlier in that sense, like Alexander Grothendick and Grigori Perelman. Many other great mathematicians, like Terence Tao, Timothy Gowers, etc. and even some working on 'big problems' like Andrew Wiles are in stable academic jobs. If his own words [1] are anything to go by, Zhang's situation is probably due to his adviser's unwillingness or inability to recommend him for good academic positions.

[1] http://nautil.us/issue/5/fame/the-twin-prime-hero

[cma]

Andrew Wiles had to do all the work in secret while releasing old papers as a cover.