What do you mean by data export? What do you want to achieve?
If it's a Story Map, go in and edit/configure the story and 'Publish' the story.
If it's a Map Tour, then it's a Web Mapping Application which is saved and then either viewed as-is or embedded in a story or other apps.
I want to have a local backup of the data I filled into a Story Map's Map tour interface.
It is both insurance from arcgis going down/ability to move my data and use it with different provider as well as being a GDPR compliance requirement.
@povilasid -- If you used ArcGIS StoryMaps to create your map tour story, you can use the Print option (in the ... menu in the header) to export the story as a PDF.
The classic Map Tour template doesn't have a print option, but you can export the data from the feature service as @DavidPike mentioned. Just go to the item page for the feature service (not the story map), and you'll see an Export Data option there.