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GeoStory is a product associated with GeoNode that preforms the equivalent of Esri Story Maps.
MapStory is a possibility. It's a platform for "playable" maps of time dependant information. I've heard it referred to as "YouTube for maps". Not exactly the same as Esri Story Maps, but there are some similarities.
You should try TimeMapper made by the OKFN (Open Knowledge Foundation).
Explore some examples like the Major Battles in the Napoleonic Wars or the US City Bankruptcies - Spending Stories to have an idea about what it does.
See also another project called StoryMap JS by the "Knight lab". It does something quite similar to ESRI or MapStory solutions.