I'm a US citizen, and recently visited my embassy abroad. I was denied entry and told to make an appointment. The first available was about 10 days away. This was for a simple document I needed signed by them. And I was leaving this country in less than 10 days. The guard who turned me away said they used to take walk-ins but not since mid-2012.
So as far as embassies go, being a citizen doesn't help too much.