Make sure you have these in your exim.conf:
hostlist relay_from_hosts = 127.0.0.1
# allows mail from localhost to be relayed to an external server
# allow "firstname.lastname@example.org" addresses.
Note that "allow_domain_literals" is NOT a recommended setting, especially on a production machine. If I understand, you are just fooling around with this, but ip addresses can easily be spoofed, turning your box into an open relay for spammers.
A better idea is to create a hostname for your box (if you haven't already) then edit/add these lines in exim.conf:
primary_hostname = your_hostname
domainlist local_domains = your_hostname
If this still doesn't work for you, you can test if the problem is with exim or not by telneting to port 25 and sending mail manually:
[me@mine]$ telnet localhost 25
Connected to localhost.
Escape character is '^]'.
220 nina.badcomputer ESMTP Exim 4.41 Wed, 08 Sep 2004 12:51:57 -0700
250 nina.badcomputer Hello localhost [127.0.0.1]
354 Enter message, ending with "." on a line by itself
This is a test email
250 OK id=1C58a9-00043g-M6
221 nina.badcomputer closing connection
Connection closed by foreign host.