For RFC compliant mailservers (RFC822 6.3, RFC1123 5.2.7, and RFC2821 4.5.1)  you are required to accept mail for postmaster for every domain you host (also accepting abuse@ all domains is considered good practice) the simple way to do this with exim is as follows:

Create a file called globalaliases in /etc with the aliases one per line in the format: - The first part is the alias at any domain, the second is the recipient address, note that the first two are local users (convention is to forward roots mail in /etc/aliases)

postmaster: root
abuse: root
somethingelse: you@yourdomain.com

In the directors part of your exim.conf file

mysystem_aliases:
driver = redirect
allow_defer
allow_fail
data = ${lookup{$local_part}lsearch{/etc/globalaliases}}
file_transport = address_file
pipe_transport = address_pipe
retry_use_local_part

You'll then have addresses that won't be interefered with by users having :fail: or :blackhole: Default Addresses.