My .02

I would say no, unless you intend to build a email system which is not meant for communicating with other email servers. The point of the domain is to uniquely identify route email to a domains' email server.  If you exclude the domain, you can only deliver the email locally because you have not included the routing information - it's equivalent to putting a letter into a mailbox without an address versus hand-delivering a letter.  Without an full email address, you can't deliver the letter.

Can you explain the problem you would like to solve?  For instance, Do you want user1@example.com to be able to send an email to simply "user5" rather than user5@example.com?  That situation could possibly be made to work  - or the user could simply use the address book, but the email would still really be addressed to user5@example.com so I don't believe I understand your intention.

But if you want to not have a domain entirely on your email server, that is probably not possible or recommended - you'd probably be better off with a bulletin board/forum if you want a mechanism for the exchange of messages which doesn't require a the format of username@domain.

Or have I entirely misinterpreted your request?