I've seen that being done around here, since the hardware you're describing really is pretty close to a beefy desktop/home server which usually spend their lives idly spinning their fans.
Rackspace Cloud for relatively cheap compute once in a blue moon
However, they noticed the downtime problem and emailed me saying that a technician had been dispatched to look at the problem. Within 15 minutes, they fixed my GRUB configuration and had my server back up. And, they didn't charge me any for the work the technician did.
It was awesome, and they immediately won my long-time business!
Disclaimer: I work for Rackspace
E3 w/ 32GB ram or dual E5 with 384GB ram.
For an instance you run only rarely the somewhat higher cost of EC2 instances compared to other providers should be fairly irrelevant. You'll find options from about 30 to 80 cents an hour.