You can register a domain name and not have a website (but be warned, the odds are that it could show anything from a simple advert for the domain registration firm right up to a Google AdWords page showing get this your main competitors, based on matching the domain name to a business category).
If you are after multiple mail addresses there are some commercial options (eg Google or Microsoft) which might be what you want, or a commercial e-mail service such as Runbox.com which can handle multiple accounts under one manager account/ bill payer.
I offer hosting, and if you want to use the Message Poster option we can perhaps chat a bit more to help you get to some solution (not from me. probably, but I am happy to ask the 'searching questions' which might help you be clearer on what solution(s) would meet your needs).
Some registration firms will give you basics for free, but multiple mail accounts on your own domain are unlikely to be free - you've also got to consider what you might want in the way of archiving any mail (for 6 years, in terms of finances, etc).
I use a number of hosting firms, so may be able to find a deal with a lower cost per year for your registration. Don't forget you may be able to register a short name .uk (unless there's a .co.uk already, as it will be 'reserved' for them).
If you have a fixed IP address on your internet connection, you could even run your own mail server, so you'd have complete control over mail, virus scanning, etc, but price would depend on number of accounts you need.