HTTP 503 is a temporary error code (“Service unavailable”) indicating iCloud servers aren't responding correctly to requests from your devices. The error typically gets resolved as GoodNotes automatically retries, but we're getting many reports of the error lingering on, causing sync failures.
This issue is not apparent to us and we've escalated the case to Apple Technical Support team for investigation. We know of other apps experiencing the same issue. 
For now, please try these suggestions.