You can tell a "story" without necessarily creating a "narrative". I think you may be hung up on believing the only form a story can have is to create a narrative.
Some images very clearly create a narrative. Here a young person has been frightened by an older person wearing a mask that is a bit spooky. The body language of both subjects and the facial features of the mask speak in a near universal human language that communicates the narrative.
Does the following image tell a story? If so, what, if any, is the narrative?
For those who have experienced a climate where leaves change colors and fall with the seasons the image might evoke memories of a particular place and time where a similar scene was experienced. At the very least for those viewers the image does tell a more generalized story of sorts: It reminds them that when the days become shorter and the temperatures become cooler the green of summer gives way to the colors of autumn. But for someone who has spent their entire life in a desert or tropical climate that does not experience the four seasons and who has not learned about the four seasons that occur in other parts of the world the image doesn't tell that story.
This is because the story is only implied, rather than being explicitly narrated, in the image. It isn't a narrative in the sense that we aren't really informed of where a particular clump of leaves on a particular tree in a particular place at a specific time became the colors we see. We're not told what color(s) they were before they became the colors in the photo. Rather it is more of a universal symbol for any leaves that change color in the fall.
Beyond that, I'm not sure that I agree that every good photograph must even "tell a story". I think the purpose of the visual arts in general, and photography as a visual art in particular, is to evoke an emotional and/or an intellectual response. It matters not whether a work of art includes a narrative or even a story at all, as long as it invokes a reaction from the viewer.
This image, in and of itself, doesn't create or illustrate a narrative. One would have to really stretch the meaning of the word story to say that it tells a story at all. What it (hopefully) does is allow the viewer to bring their own story to this fairly abstract image of fireworks and give the image meaning that draws from each viewer's own experience. A botanist who cultivates flowers and dandelions may have an entirely different interpretation and reaction than a combat veteran who has survived anti-aircraft flak over Europe during the Second World War.
On the other hand, if one were to place a title that references an existing cultural or literary narrative beneath the above image then the image could illustrate that existing narrative, at least for those who recognize the reference(s) in the title. Let's call this one "Michael - Lucifer - Gabriel". Now something that was fairly abstract visually has been assigned a meaning and those who recognize the literary reference¹ may began to see a relationship between certain forms in the image and the narrative of Lucifer's fall from heaven.
Whether narrative or not, some works will contain culturally conditioned elements such that the emotional response from most viewers will be similar to most other viewers from the same culture. Other works may be more abstract and invoke different types of responses from different observers based on their own life experiences.
¹ Michael, Lucifer, and Gabriel are the traditional names in the Judeo/Christian tradition given to the three archangels of heaven before one of them, Lucifer, fell and was banished from heaven, along with one third of all of the angels in heaven who followed him, for daring to consider himself as equal to God.
From the comments:
Could you expand on the idea of stories that aren't narrative in nature? The two words are often synonymous, especially in the context of tell a story.
And a response from another user:
An image is usually only showing a moment. Unless it depicts the story as it unfolds and shows the "action", it doesn't narrate the story. It might show you a particular slice in time of the story. Your imagination and other sources of information create the story, but the image alone is rarely telling it all.
I think the distinction between "narrative" as a subset of the more general "story" is somewhere in between the two poles expressed in the two comments cited. One extreme sees "narrative" and "story" as having near identical meaning. The other sees "narrative" as applicable only when everything in the story is visually made explicit.
In reality it seems to me to be more of a gradual continuum. In much the same way that we use the concept of depth of field there may not be a clear cut "line" that divides a more complete and explicitly expressed "narrative" from a more general "story" that depends on the observer to fill in the details with their own experience. The three examples I've added above move from more explicit narrative to more generalized "story". In any case I think we would all pretty well agree that all narratives are also stories, but perhaps not all stories are narratives, at least not explicitly so with regard to the visual arts.