There are a few different types of cloud computing.
I don't know about AWS on this but Azure has the concept of "Web Sites" which lets you host it in a load balanced server farm. Your site will literally never go down.
I don't know about Azure, but that sounds like Platform as a Service. AWS is Infrastructure as a Service.
That means you have access to infrastructure (servers, load balancers etc.) on demand. But you are responsible for all software on the servers, including the operating system. The hardware is taken care of by Amazon, and you can start and stop servers when you want, provision and attach disks in a matter of minutes with a few clicks and back them up instantly whenever you want.
But although the hardware is taken care of by Amazon, the servers can and do fail, and when this happens, it's mostly down to you, not Amazon, to fix things. You can choose which zone to start a server in, and each zone is in a different data centre, so to make it reliable, you need your application hosted by multiple servers in multiple zones.
As NathanE said, it's not really designed for a single small site.