Is it impossible to have a decentralized service using email addresses as login?
The core of your question is "Is there an attribute that an 'owner' of userA@example.com can be proven to have, and which no on else can be proven to have?" This heavily depends on the security of the email system, which, in general, is quite poor. Another central question is what "registration" even means. Presumably, it means "proving" one's identity, but proving it to whom, and how do others know that it has been proven? There has to be some process by which the "network" verifies identity, and then there has to be some process by which users verifies that the network has verified identity. Fully answering your question requires a precise formulation of what those processes are.

Ultimately, the only two possible unique attributes that an "owner" could have is the ability to send email, and the ability to receive/read email. Any other attributes, such as the ability to sign with a private key (asymmetric encryption) would mean that you're not basing it solely on email anymore. At that point, it's like asking "Is it possible to open a bottle with a spoon, if you also have a bottle opener?"

If you are confident on being able to detect email spoofing, then you can just have someone choose a private and public key, send their public key from their email to the nodes, and then sign everything with their private key. 

If you're not confident that you can detect email spoofing, but you are confident that it is possible to send a message in such a way that only the "owner" of an email address can read it, then it's impossible for any single message to generate any attribute unique to the new user; any information sent in the email, the sending node also has. So you will have to have "the network" in general send multiple messages from multiple nodes, each with their own token.