Is it possible to use Docker to separate web sites for users?

Yes, it is possible. What you need to do is providing several 80 ports. one for each URLs. You can do this using, e.g. Virtual Host of Apache running on Docker host server.

1. Set DNS CNAME.
2. Run docker instances and map their port 80 to port, say, 12345~12347 of the docker host.
3. Run Apache server on docker host and set a Virtual Host for each URL and set ProxyPass and ProxyPassReverse to localhost:12345 which is one of your docker instances.

Apache config file will look like this:

<VirtualHost *:80>
ServerName www.somewebsite.com
  <Proxy *>
    Allow from localhost
  </Proxy>
  ProxyPass        / http://local.hostname.ofDockerHost:12345/
  ProxyPassReverse / http://local.hostname.ofDockerHost:12345/
</VirtualHost>

It is possible. You may use apache (or better yet, haproxy, nginx or varnish, that may be more efficient than apache for just that redirection task) in the main server, to redirect to the apache ports of each container.

But, depending on the sites you run there (and their apache configurations), it may require far more memory than using a single central apache with virtualhosts, specially if you have modules (i.e. php) that require a lot of RAM.