Archive your email, contacts, and calendars before it's too late
The upshot is that you'd like to make copies of this before it vanishes. How do you do it Let's start with Mail.
The trick to grabbing your old email is making an archive of your selected inbox or account. In this case, I have a bunch of mail stored in Mail's inbox for a Google account that I'll say is my work email account. To make an archive of it I just select Export Mailbox from the Mailbox menu. (Or I can just right-click on my inbox and choose this same command).
When I do this, Mail will set about creating mbox files for the mailboxes within this inbox. At a later date I can then import this mbox into Mail or another email client and have access to the email and its attachments. Note that if you have a lot of messages and attachments it can take quite awhile to do this.
Contacts held on the company servers are just as easy to archive. Launch Contacts, choose File > Export, and choose Contacts Archive. This will take all your contacts and shove them into a single archive, which you can later import into Contacts when IT disallows your access to the company servers.
Calendars work similarly. Just launch the Calendar app and you can choose to export a calendar archive that contains all your events, or select a particular calendar and choose to export just that one.
But what if you do this kind of thing through a web client. If you're using Exchange or Office 365 you're out of luck--Microsoft doesn't provide the means for archiving your stuff from within its web client. You'll have to use a local email client such as Mail or Outlook and follow the steps I offered earlier.
Google does provide a solution, however. Go to www.google.com/settings/takeout and you'll see a page like this where you can choose to download all the stuff you've shoved into Google--including mail, calendars, contacts, and files on your Google Drive. Depending on how much stuff you have, it can take a long time to create the archive. Google will email you when it's complete and ready for download.
They say you can't take it with you. Now you know that's not entirely true.