The actual zonefile is hosted by a group of volunteer organizations. The variety, and the use of anycast means that the root zone is fairly stable and as reliable as any other distributed system (i.e. cloud). You can even download the root zone and run it locally if you want.
Handshake is interesting, but it conflates DNS and the CA ecosystem. They are different and used for different things. Most of the big issues around DNS isn't about the protocol, but the policies and legal issues around domain names and their ownership.
But it’s a single root, and it is administered by ICANN directly who decide who can be a root name operator. So there is central control there.
building some sort of overlay network on top of the internet has been done before (.onion anyone?).
How would this improve upon such a scheme for example? I really don't think it adds anything but the word 'blockchain' to the equation, and i honestly don't think that adds anything in the way of security.
additionally, to revoke ones domain name to say, resolve to another host, changing hosting provider for example, would seem impossible in an immutable blockchain. Would that result in forks or in namespace pollution?