The default EXIM config does not allow relaying.
# Insist that any other recipient address that we accept is either in one of
# our local domains, or is in a domain for which we explicitly allow
# relaying. Any other domain is rejected as being unacceptable for relaying.
require message = relay not permitted
domains = +local_domains : +relay_to_domains
If you remove the above argument (line 455 in my /etc/exim/exim.conf) then open relaying will be permitted through EXIM, which is bad. However, you can give it shot for testing purposes. Understand first though that ANYONE will be able to relay mail through your MTA at that point - eventually SPAM will disseminate from your server, your domain will end up on blacklists, etc. if you don't restore it shortly.
In order to make EXIM function like a modern relay server you must read up on the topic of authentication. EXIM's default config will permit relaying for authenticated email clients. Here's the official book: http://www.amazon.com/Exim-SMTP-Mail-Server-Official/dp/0954452976
Are you, or do you intend to use LDAP to store usernames and passwords - as it pertains to authenticating your email users?
Here's also a pretty good article. It should give you a holistic idea of what it takes to setup an EXIM MTA based email server - with LDAP I should add. The howto is Debian specific: http://www.howtoforge.com/setting-up-a-mail-server-using-exim4-clamav-dovecot-spamassassin-and-many-more-on-debian-p2