This problem is documented in the FAQ at exim.org. You need to set localhost as a recognised domain for mail to the machine you are on to be addressed to, otherwise the relaying controls deny you to relay the the localhost. You can either set this with eximconfig, or edit the config file (I can't remember the line that you need to edit, sorry but you'll have look at the FAQ or wait till I get home from work)
Regards,
Andrew Clark.
