I have a box running at DigitalOcean with Ubuntu 12.04
It's rare that my system will boot or takes forever - normally it just hangs / chokes. A reboot from a remote shell used to be pretty quick and I could reconnect within 10 seconds or so. Here's screens shots of where it gets to:
I have been told that such may happen after a kernel upgrade if their "hypervisor" doesn't have that kernel and that it may require manual setting from DO's support.
I had them mount their ArchLinux recovery disk and checked out the system's /lib/modules: I've been told these are the supported kernels that Ubuntu can run with.
I set my kernel to: ubuntu 12.04 x64 vmlinux-3.8.0-42-generic From DO's droplet settings.
I'm still getting the above hangs and some of the images above were after such.