I am using RDS to run a postgresql server (9.6.3) and this morning, a backup was automatically kicked off. It is still going 6 hours later which seems absurd. The database is not that big (~ 600 GB), and as far as I can tell, this is the first time i've experienced this problem. The machine is relatively beefy (db.m4.2xlarge), so it seems like these backups should take a lot less than 6 hours. I am also surprised by the fact that a backup would be kicked off at 5:30 AM, which seems awfully close to standard biz hours.
Any ideas?
You scheduled the 5:30 AM backup window. Amazon didn't randomly kick it off at that time. Look at your RDS instance's settings and you will see a backup window that was defined when you created the instance.
An RDS backup is like an EBS snapshot, and it shouldn't be reliant on the CPU of the server at all. It should also not affect server performance at all.
You should look into migrating to Amazon Aurora now that the PostreSQL version is out of beta. Among other benefits, you will get extremely fast snapshot creation with Aurora.
Sometimes things like this become "stuck" due to an issue behind the scenes. If that happens all you can do is open a support ticket with AWS to get it fixed.