Yes it is completely possible to host websites on AWS in 2 ways:
- Easy - S3 (Simple Storage Solution) is a bucket storage solution that lets you serve static content e.g. images but has recently been upgraded so you can use it to host flat .html files and your site will get served by a default Apache installation with very little configuration on your part (but also little control).
- Trickier - You can use EC2 (Elastic Compute Cloud) and create a virtual Linux instance then install Apache/NGinx (or whatever) on that to give you complete control over serving whatever/however you want. You use Security Groups to enable/disable ports for individual machines or groups of them.