OUT OF HARD DRIVE DISC SPACE

Be sure you are using GUID for the partition scheme and formatting the drive using Mac OS Extended, Journaled. Do not use APFS on the backup drive if you are using Time Machine.

Run First Aid on the internal drive and the backup drive.

Repair the Drive for El Capitan or Later

1. Restart the computer. Immediately, at or before the chime hold down the Command and R keys until the Utility Menu appears.
2. Select Disk Utility and press the Continue button.
3. Then select the indented (usually, Macintosh HD) volume entry from the side list. 
4. Click on the First Aid tab in the toolbar and wait for the Done button to activate. Click on it, then quit Disk Utility.
5. Select Restart from the Apple menu.

If the problem is with your internal drive then verify you have a problem by trying to copy one file to the backup drive using the backup utility you were using when you got the error. What error do you get, if any, for a single file copy?

EtreCheck is a simple little app to display the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to help people help you with your Mac.

http://www.etresoft.com/etrecheck

Use the Note tool on the bottom of this editor's toolbar, as shown in the image, to copy and paste the output from EtreCheck.

Most likely you've run out of space on your backup drive. Deleting files on your source drive won't help.

Is the drive formatted for HFS+ (Mac OS Extended)