: How could the disregard for both plot and dialogue tell the story? I want people to understand what I write, but I do not want to use conventions such as plot or dialogue to tell a story.

There are as many ways to tell a story as there are people to tell. And, how people tells stories has evolved continuously from the dawn of humanity and will continue to grow and change until the last human dies.

The conventions of plot and dialogue and rising action and et cetera are observations about qualities told by successful story tellers. They aren’t the only means that can exist to effectively tell your story. You may have worked out a brilliant new structure and mode of storytelling, or you might be horrifically bad at it.

But, its your story to tell the way that makes sense to you. And, you need to decide, based on people’s responses to your stories, if you are successfully telling your story. Because that is the goal of storytelling, to share your story, your art, and have other people understand, and possible]y enjoy, but mostly engage with and pay attention to it from beginning to end.

Oh, yeah, one more thing, you just can tell the stories to your parents and your friends. They like you, and will tell you what you want to hear. You have to tell it to people who don’t give a damn about you, but will still listen from start to end.

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First of all, I wasn't so clear on how you're writing your story and what you're calling a plot. There are many definitions of a plot, most of which you probably want to include in your story.
Take this one:

Plot is about forward movement toward a specific end.

And:

Plot: the plan, scheme, or main story of a literary or dramatic work,
as a play, novel, or short story.

And this one: Plot is the story's sequence of events.

While it is possible to write a story without a plot, it's not necessarily suggested. Your story may lose people's interest, sound like it has no point, or get sidetracked on unrelated points and confuse people.

Once you're writing about characters and events it would seem to me that you have a plot. Because, in essence, plot is the story, and is hard to avoid (if you're trying to write a story).

If you have a theme or central message, then you could theoretically get those across without a set sequence of events. It would all depend on what you're trying to write. Are you writing a story? A poem? Random descriptions like @NofP 's last quote? A string of thoughts?

There are other ways to write non-conventional stories, besides cutting the plot, which may be worth exploring. To give your stories more of a story element and help them make sense, here's what I suggest:

Know what happens in the story. (This would be considered the plot element). What changes throughout the story?
Know the message that you want to give across. What's the story's theme?
Know who your characters are and what they are adding to your story. Are they all there for a reason?
Omit all points that don't relate directly to the story's central plot/message. Otherwise it might make no sense and all seem like unrelated points.

Dialogue can be avoided at large by gestures, letters/emails/texts and a lot of internal dialogue, though it sounds like you needed less help in this area.
Hope this helps.

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I think you are not writing a story, you are writing a vignette that captures a moment; this is more akin to poetry or a painting or a song or photography, those all (aim to) capture a feeling, emotion, or dramatic moment. (I am presuming this is fiction, and not an academic essay detailing some process or proof.)

This is still art, it is still writing, it just isn't a story.

A story (IMO, and opinions vary) is a narrative about a struggle that is won or lost. Typically by some main character, with a stake in the outcome, often with a villain. A story has conflicts, the MC wants something to happen (or not happen) and is met with resistance, or confusion about how to bring it about, etc.

I would call what you write a vignette, or essay, or sketch. To get better at that, I would focus more precisely on what moment or feeling you are trying to convey, and ensure every line contributes to that, and nothing else, don't cross signals. Study how poetry works (I haven't), or read fiction not for the plots, but for the parts where an author describes feelings, or settings, and tries to set an atmosphere. That is the kind of thing you are trying to capture also, and you can analyze sentence structure, word choices, the mix of sentence lengths and tense to see how they accomplish this, and try to emulate that, or use it to edit and improve your existing vignettes.

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A knowledgeable Writing.SE user once said you could write fifty thousand times the word 'meow' and call it a novel. Such a piece of writing would hardly be considered an account of anything, even less so a 'story'.

If we dial back from the extreme, you could consider some random sentences, like the one that my computer can produce.

The cow was being held captive for a week to make it easier to find a spot in my heart. The singer wrote me that the film would wait. The first of these, the second of those: he has a very different story but it is not the only reason to be skeptical that the world is actually flat.

No dialogue. Definitively no plot. It is a story in the broad sense of an account of people and/or events, but is it an interesting story? Most likely not.

When someone asks about developing a character, the typical answer on Writing.SE is to make them relatable. Give them something to connect to the reader, be it a goal, a struggle, a particular detail or situation. The same goes to just anything in writing: readers will read what they find interesting, and giving them something they can connect to will grab their interest.

Let's consider an even less drastic example, you could consider a text that is just the description of a landscape. A snapshot of a particular moment in time.

Some hills, a flock of sheep grazing on the side of a barn. A family walking by. The youngest pointing at the shadow of a cherry tree. A car dashing on the road. In the sky a hawk is circling over the fields.

This has no dialogue, and no plot. This is however a story. And, in contrast to what my computer writes, you may now guess what is going on, and even get a mental image of the scene. This may even connect with the reader.

You could improve the connection by using a better style, by evoking more profound images or thoughts. By raising questions in the reader that will push them to continue reading in order to find an answer. If I were to edit the landscape description to include these elements, it could become:

Some hills protrude unexpectedly from the plains around Garning. On the soft slopes, a flock of sheep grazes by the side of an abandoned barn. A family is walking by. It is four of them, with faces covered in dust and sweat. Their legs are shaking, and the oldest staggers every few step, like a sick lamb who does not want to be left behind. The youngest is far ahead, almost hidden amidst the sheep, and turns back his head pointing at the shadow looming from under a cherry tree. Isn't it the perfect place for a final rest? He may be already screaming from his cracked lips, but the words are covered by the roaring of a car dashing on the road that cuts the plain. In the sky a thin-looking hawk is circling over the sunburnt fields.

Details. Picture the image before you write it. Imagine the circumstances. Perhaps the family got lost. Perhaps the hills are a magical place outside of time. Maybe the sheep are aliens sent to kidnap humans, and this family is walking right into the trap. You don't need to tell the plot, or even have a dialogue to create a single scene that contains all these elements. The better you can convey these images the more interesting for your readers to read.

You could even have a more abstract theme. No plot, no dialogue, no characters, and (I tried) not even a bucolic description of the countryside.

The distant Moon is large, round like a screaming face, howling in the night, over the woods. Around here the infinite sparkling eyes of the stars, watching over the dark leaves. Not even the sense of loneliness seems to break through the ticking of time. For time is but one of many directions, and loneliness is immobile.

My suggestion to tell a compelling story:

define a skeleton of the elements of your story
expand on them with as much detail as you can image
cut back to the size of a readable text.

And, stay true to the advice of showing and not telling. You don't need a plot, nor dialogue, not even characters for that matters. All you need is a good idea that resonates with the reader, and practice a good writing style.

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