We have a server, we are using this for backups and other cronjobs. I would like to send mail to the users when one of their cronjobs or backups has ended. Is it possible this if I don't have a domain name?
Yes it is possible to send emails without having a domain name, if you currently make use of Gmail you can use your existing Gmail account to send emails as desired. This is a great tutorial which can help you configure it all:
UPDATE for link:
This is one example of how to do it, you can use other email accounts it is not restricted to Gmail, but I believe this to be the most common.
Is it possible this if I don't have a domain name?
The sort answer is Yes. You don't need a domain name to send email and my servers regularly send me emails (through
postfix - you do need a MTA) about things from their hostname and my computer does the same. I'm constantly getting emails from
bert is my hostname).
The "problem" is that the emails might be blocked without a fully-qualified domain name because lots of anti-spam stuff relies on FQDNs to score emails' spamminess. I personally don't have any problems sending mail to myself on Gmail but you might have problems with Gmail or other providers.
The most professional solution would be to get a proper domain and set things up but if don't want to stretch the $10 or so, you can get a free email account with Google, Hotmail, etc and most of them will let you send email via SMTP. And then
postfix can relay those emails
The email sending is solved here: How to send mail from the command line?
The easiest way to send an email without the complexity is to use the sendemail package. Assuming your server can see some kind of SMTP account, it's a breeze.