For me, I had to run the patch a few times. Due to randomness (or maybe just sloppy closed source coding) SharePoint CUs tend to fail. The good thing, is that they either succeed 100% or fail completely (leaving SharePoint more or less in an OK state. Luckily for me, I did not have any issues running the product configuration wizard. I've seen and heard of instances where product configuration wizard fails, but its usually a clean up task that fails and isnt a big deal.
After I updated the March 2013 PU, I ran into a 503 unavailable for central admin and my site collections. Instead of finding an immediate resolution, I plowed forth and installed the June 2013 CU with success. Unfortunately after the database upgrade and a SharePoint server reboot, I was still getting a 503 error when trying to access SharePoint.
After a bit of googling, I found a working solution to this particular problem. Load up IIS Manager and head to the Server Application Pools. As per this post and for me, all of my Application Pools were stopped. Without hesitation, I started every stopped pool, restarted IIS and once again was able to access Central Admin and my Site Collections.