Is it possible to use Amazon CloudFront as a hosting provider (linking it to a domain, etc) or is it only meant for content hosting?
The "hosting" provider is more likely S3. You can easily host your web site there. See instructions here: http://docs.aws.amazon.com/AmazonS3/latest/dev/WebsiteHosting.html
Amazon CloudFront is a CDN service, which makes serving content faster for your end users as the content is available from edge points closer to the end users. You can create a distribution of your web site (from example S3 as explained above), but the content must arrive from an origin (as S3). See instructions here: http://docs.aws.amazon.com/AmazonCloudFront/latest/DeveloperGuide/GettingStarted.html