You would like to have an email server connected to your intranet, to send and receive intraoffice email, without making an internet connection.
If you happen to have a PC around with 256MB RAM and 20GB hard drive, Linux like Fedora Core with Sendmail and IMAP support may be the answer. Each user has a local account where the mail is stored. You may have an intranet domain, or the email address is simply the username, without the @ sign.
The Sendmail configuration needs to be fine tuned so that it does not relay mail to outside servers.
Mail clients like Outlook or Mozilla Thunderbird will be able to access the mailbox as usual. If you have restricted the allowed DNS requests in the LAN to the internal servers, or even better, if you have a locked-down desktop, the kids will not be able to change the settings and access DNS and mail servers outside your LAN.