SEOUL—Choi Young-ju earned her law degree three years ago from a South Korean college. She still hasn’t found a job.

For two years, Ms. Choi has been trying to pass the government’s civil-service exam to get hired as a police officer—if she can be one of the less than 1% of female applicants accepted to the force. She has failed three times, a familiar tale among a generation overflowing with college graduates competing for too few jobs.

...