Odds of Getting In
While the qualifications to work at the post office are less "qualifications," per se, and more "do you eat, sleep, and breathe regularly?" that doesn't mean you're automatically in. In case you haven't noticed the rise of email (or as Grandpa Charlie calls it, "e-what now?"), the postal service is going through a bit of decline at the moment. That inevitably means fewer jobs, which means more competition for said fewer jobs.
When there are positions available, USPS will administer tests of basic memory and communication skills to all who wish to apply. After grading, the hiring process starts from the top score and moves down. Do well, and you'll be wearing that dark blue blazer in no time.