I'd like to completely shut down my ARK server (and stop paying for it) for a few months as people are taking a break from the game. However, when we all return, I'd like to come back to the map/progress we had before. Am I able to do this by making a backup and then downloading that file, so I can upload it later when I open a 'new' server and start paying again? I can't see anywhere I would upload a manually downloaded backup file.

Or, do I have to go through all the steps listed on the SFTP page like downloading FileZilla, accessing the server files through there, and making a copy of all those files to store on my harddrive (and then copy them back into the remote server when I start a new server)?

You can make the backup and then download the backup. To restore the backup, as your account will have been temporarily closed, you'll need to manually upload your backup after extracting it on your local computer.

Another thing to note:
Backups are just your map file, your player files, and tribe data. They do not include your settings. So you will need to re-configure your settings when you come back.

Perfect, just what I was hoping to hear. WHERE do I upload the backup (when I return)? I can't find anywhere to do that, which is what made me wonder if it was even possible.

As was mentioned, this would be a manual upload of your backup. 

Meaning you'll have to decompress the backup on your local computer, then connect to the server (while stopped) via SFTP, then navigate to the appropriate folders for each of the files contained within your backup.

Depending on your backup and it's contents, the folders and directories will vary. 

In short: It will be as if you're migrating to ARKServers.io from another host.

Instructions for manual migration can be found here:

https://arkservers.io/help/migrate

Oh ok - so I'm not downloading the 'backup' file from my control panel, the file ending in .tar.gz. I'm connecting to the remote host and copying all the game files from there, and saving those on my PC somewhere, to later upload.

Though you mentioned being able to use that .tar.gz file as my saved backup, and then decompress it before uploading it to the remote host - is there a specific program I need for that? I didn't know that .gz was a regular file type that could be decompressed.

I think 7zip will do it, though .tar.gz is a Linux native compression format.

It's also a VERY good idea to copy down your settings from the ini in a human readable format, since you won't be able to upload the ini files here.