You can get really close by using the one click Dokku deploy on DigitalOcean. Technically it is your own VM, but in practice you just set your git remote to that IP and use it like you would a Heroku instance. It even supports the Heroku build packs!
The blog post we're all commenting describes the point of heroku as to lowering the barrier to entry for deployment. It's about the number of things that you don't have to think about. As a sibling comment says in Google App Engine you also don't have to run a server, but it's much more complicated to use (I ran a startup on it for a while).
