Is it possible to use Amazon web service to host a Rails application?
Yes, you should be able to set that up in several different ways on AWS.
Since you can configure an EC-2 instance much like a dedicated server, you can install any software you want and configure it to your needs.
There are some nice ruby tools here:
Also 37 signals is on AWS – http://aws.amazon.com/solutions/case-studies/37signals/
I recommend a simple Amazon Linux AMI Micro instance EBS backed to get started on the free tier.