You haven't told us what your preferred email client is! So there is a limit to what can be answerered.
There is a good answer for Thunderbird in AskUbuntu: https://askubuntu.com/questions/1916/how-can-i-access-system-mail-in-var-mail-via-thunderbird.
Basically, Thunderbird supports
/var/mail directly and you can add a folder there as a "Unix Mailspool" when adding a new account.
If you use a different client, let us know so that we can provide a better answer. Please also tell us what OS/Distro you are using.
UPDATE: As mentioned in the comments. The files in
/var/mail generally use the MBox format which may help others find suitable clients for their platform that can read them.
Also, as mentioned below, the alternative way to make the mailboxes available to any mail client is to run a mail server such as Dovecot or Courier which will present the files using a standard protocol such as IMAP or POP3.
UPDATE2: @blacklight-shining has updated me and said that there are 1/2 dozen boxes that need monitoring so I've recommended that the best way forward is to use SSHFS (and FUSE for better integration) on one of the boxes to make the folders from the other boxes available in a single place. Then, on that single box (a Debian server), to run up a Dovecot server with just IMAPS (IMAP4 over SSL). No SMTP server is needed as there is no need to send mail only to access the mailboxes which are recieving system mail.