Lars -- If the table needs to be a live view of data it would need to be hosted on the web somewhere as a viewable table, so yes Google Sheets or Office 365 would work for that. But, as you pointed out, for the table to be displayed inside a story map it would need to be shared publicly. Story Maps don't support private embedded content because browsers block showing a login screen inside another page (that is, inside a story map) since it violates web security practices.
You could download the code for a story map app and customize it to provide a login experience for the site you want to use to host the private spreadsheets and host your customized version of the app on your own web server. The reader would be prompted to login before the story map loads and would pass the credentials to the site of the embedded content.
Owen Evans
Lead Product Engineer | StoryMaps