You have to add the "storage" permission in your manifest.json file, i.e.:

...
  "permissions": [
    "storage"
  ],
...

For more information, see: https://developer.chrome.com/extensions/storage


RELOAD THE EXTENSION

I had the "permissions" key added in my manifest file but still I struggled to get this fixed.

After adding the permission:-

"permissions": [
    "storage"
 ]

Goto your extension using: chrome://extensions/ & click the Reload button:-

enter image description here


If someone was facing this issue on Firefox, please note, that it is not supported yet:

https://developer.mozilla.org/en-US/Add-ons/WebExtensions/API/storage#Chrome_incompatibilities

For my purposes it was sufficient to replace chrome.storage.sync by chrome.storage.local.

Regarding the Firefox implementation state it might be worth to look also here from time to time:

https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/WebExtensions/Browser_support_for_JavaScript_APIs#storage


RESTART THE DEV SERVER

Despite adding the permission and reloading the extension, I sync was still undefined.

Turned out I needed to restart the dev server for the new manifest to picked up. Which makes sense since this file was not being watched. Doh!