Yes, in BigPicture, you can create such a structure even though stories and tasks are on the same level in Jira.
Please note that Task Structure can be created based on both build-in task attributes (e.g., Epic link, Component, Version, Sprint) as well as links.
To achieve the expected outcome, I recommend linking story and task by, e.g., parent/child link, and then use this link as a structure builder in Program Configuration >> Tasks >> Task Structure.
Would you please let me know if you have further questions?
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