Odds of Getting In
According to the Federal Government, there are twenty-two different groups of public administrators, and each of those has a wide range of positions for professional PAs. Such positions are available at all levels of government, in countless different non-profit organizations, and in the private sector consulting businesses on dealing with the public.
If you're qualified and make a good impression on whoever does the hiring, you shouldn't have much of an issue opening that door to city hall―figuratively or literally.
One caveat? Over 44,000 students a year graduate with a degree in public administration, and that's just the ones with bachelor's degrees (source).
Of course, since 45% of those go back to get that masters instead of jumping right in to a job, there's not as much competition as you'd expect right out of the gate, and you could always go earn that higher degree for the extra cash that comes with it. You actually have a lot of choice in when you decide to finally leap into the "real world."
As long as you're not too picky, that is. Hey, someone has to be in charge of the sewer system.