When you export them, you should get a normal “Save” dialog box that will prompt you for the location where you want to save them.
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I do not think that is true. Browsing a recent TM backup, I see content in both ~/Library/Application Support/AddressBook and ~/Library/Calendars.
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I have an iMac, an iPad, & an iPhone; I use iCloud on all of them (Mail, Contacts, Calendar, & Notes). Usually everything works correctly & all stay in sync. I had a problem several months ago, don’t remember whether it was w/Contacts or Calendar; something wasn’t syncing correctly.
Called Apple & eventually figured out how to fix the problem, after being moved to a Senior Advisor. I was told that while iCloud usually works seamlessly for keeping the Apple apps in sync on the various devices, occasionally a problem will occur. I was told the best way to try to avoid future problems was to do manual backups (exports) periodically for Calendar & Contacts.
For Contacts, there are 3 backup/export options (if using a Mac): Export vCard (.vcf), Export Contacts Archive (.abbu), & Export as PDF. You can also use Export vCard from the iCloud site. I do all of those Backup/Export options every few weeks. I was told to use the .vcf version if I needed to restore Contacts. To double check that everything was restored, I should open Contacts after the restore & compare it to the entries in the PDF version.
For Calendar, to export/backup all Calendars, there’s only Export Calendar Archive (on a Mac); it creates an .icbu file that can be used to restore all. No option on iCloud site for export/backup.
Every time I backup/export Contacts, I also backup/export Calendar.
It takes several minutes & a bit of effort to do all this every few weeks. But I decided it was worth to have my own work around if another iCloud glitch occurred.
Granted, it’s possible I’ll still have a few entries missing if a glitch occurs, depending upon when I did the last backups/exports. But at least I’ll have the bulk of my data.
It’s possible something could have happened (w/Apple or iCloud) during the last several months that makes these options not quite up-to-date. But I’ll cross that bridge if/when it occurs; for now I feel pretty well covered.
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Thanks for that info on backing up Contacts and Calendar, @sf.ross.
Do you have any similar guidance for Mail or Notes?
Don’t have a solution for Mail. Here’s what I do for Notes:
Each Note can be exported (backed up) individually. When you’re in a Note, go to File-Export As-PDF. Decide what folder you want to use to store the backup, & export it to that folder. I created a folder named Notes. 
I periodically export each Note to that folder; each Note has the same name it has in Notes. My (created) Notes folder has PDF versions of each Note in the Notes app.
If there’s ever a problem in the Notes app & 1 or more of the Notes “disappear” or isn’t synced correctly between my Mac, iPhone, & iPad, I can recreate the Note(s) using the PDF backups/exports.
Hope that helps. If there’s ever a problem w/Mail, don’t know what I’ll do. 
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I just checked my TM backups; they don’t show any Library folders named AddressBook, Calendars, or even Contacts. I’ve looked in ~/Library and in ~/Library/Application Support and can’t find any of those. If I remember correctly, when Apple Support told me how to backup Calendars & Contacts, it was specifically because Time Machine doesn’t back up anything for either of those apps.
You’re correct, backups/exports of Contacts do not contain the contacts groups. Learned that when Apple Support explained to me about backing up Contacts. Since that time, when I add a new contact in Contacts, in the Notes field for the contact, I list whatever Contact group(s) that entry has been added. The Notes field is included in any backup/export so if I have to import entries into Contacts from a backup, I can put everything “back” where it belongs.
A bit extra work but worth the effort in case I need to import from a backup.
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Thanks for the tip @sf.ross !
I have a family member who has lots and lots of Notes, often with multiple inserts of pdfs and jpgs inside them. I don’t use Notes, so am wondering if this procedure will produce a pdf that includes all the ‘attachments’ inside a Note…?
Also if someone has hundreds of Notes, it would be handy to batch this. Can you select a bunch of notes and do the same Export command?
Perhaps an Automator/Shortcuts/script could do this at a keyboard command, or context menu selection, hm…
I’ve been trying out two email archiving products recently for a massive (for me) digital archiving project: Mail Steward and Mail Archiver X. The latter is discounted for TidBITS members. Give them a look for keeping email available outside the Mail app itself.
Don’t know what happens to attachments in Notes when the Note is exported/backed up. If the contents of the attachment “live” in the Note, my guess is it would be included in the export/backup. If only a link to the attachment is in the Note, don’t know if the link will remain live.
I have pasted graphics into a Note & the graphics have survived the export/backup process.
I also don’t know if one can backup/export all or multiple Notes at the same time; I’ve never tried. I have about a dozen Notes, some rarely change, some change on a daily basis, so I backup/export each one when a significant change has been made. Hopefully there is a way to back up all at once.
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Exporter is an app (free with in-app purchase available) to export Notes.
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Interesting. The data is definitely there on my machine, inside the two specific directories I mentioned earlier. However, the fact that they exist may be a moot point if you use iCloud. Apple specifically warns about trying to restore Contacts from an Archive (which by extension I think would include a TM backup) “while connected to the internet and signed in to iCloud”. I think the problem is that in the iCloud-enabled scenario, Apple considers iCloud to be the “real” copy of your Contacts. So if you restore from a TM backup locally, it throws a monkey wrench into the process. Is it possible Apple Support told you that iCloud Backup doesn’t back up your Calendars and Contacts? That’s absolutely true, for the simple reason that the iCloud copy of both is already there.
On the other hand, if we’re talking about “On My Mac” files, then the TM backup could be a legitimate restore source, since iCloud is not in the picture.
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Yes, I do use iCloud. I never intend to restore from an iCloud backup, as if I need to restore, it would likely be due to an iCloud issue. That’s why I run the Contacts and Calendar backups on my Mac every few weeks. If I needed to restore Contacts or Calendars, I would use the backups I created on my Mac.
I’m certain Apple told me that TM didn’t create backups for Contacts or Calendars, that’s why we spent so long on the phone making sure I understood how to create my own backups. Maybe they were only referring to my situation - using iCloud - but it appeared they were speaking about Contacts & Calendar in general. It didn’t occur to me to question whether the advice was iCloud specific or Apple in general; I just understood what they were saying was what was going to work for me. Sorry if this created any confusion for others.
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