The name registration is done with a blockchain (ring-nameservice) and a DHT (OpenDHT). You can find the information on the wiki in their Tuleap instance.
Seems to be Ethereum... ouch. You'd've hoped people would've come up with a better way of solving Zooko's Triangle than forcing everybody to store an ever-increasing, unprunable amount of data on their devices - that also requires massive amounts of data to be downloaded, wherever you are, even over mobile networks.
In practice, even DNS-based aliases are a better idea, given the tradeoffs.
Ethereum might not have been the best choice, but there are ways they could migrate to another GNU Ring-specific blockchain (therefore a smaller blockchain; there are only so many people who are going to use Ring and they will only want to have so many separate identities) if the Ethereum blockchain really became unmanageably large.
Blockchains grow unmanageably large over time by definition, no matter how application-specific you make them, unfortunately. Ethereum isn't the problem - blockchains are.
