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.