Most stories follow this simple rule: Hardship occurs to main character and main character attempts to get through hardship. Character either gets through hardship or other hardship happens and plot ends with main character "hanging off a cliff" (cliffhanger)
But what if nothing at all bad happens to the character and only good things happen to the character? What if there are no hardships? Would my reader be disappointed? Would I be hated on? Could it even be called a story?
The last question is easy. Would it be a story? Certainly. If there is any plot at all, if anything happens, it is a "story".
The bigger question is, Would it be an interesting story?
Many children's books are like this. A child or an animal wanders around and encounters hopefully interesting things. The end.
I recently moved to another country. Before I moved I watched a number of videos on the Internet where the narrator talked about his visit to the country and things he encountered here. While they were all true stories -- or at least claimed to be :-) -- none involved any life-threatening danger. Some talked about obstacles the narrator had to face, like "how do I move my money from my home country to here?" or "where do I go to renew my visa?" But many didn't really deal with any problems, or at least didn't present them as problems. They just talked about, "Here are interestings things I encountered" and "Here's how this thing works in this country."
In general, I'd say that to be interesting a story should have the hero face some obstacle or conflict. But not necessarily.
There are stories without obstacles or hardship or conflict. Many picture books for children are like that. They just tell of a character experiencing something and getting to know a benign world. Also some non-fiction works like that. There aren't any hardships or conflicts in most cook books or travel guides. But in novels for adults the absence of any antagonistic force is rare. A famous example is Der Stechlin by Theodor Fontane (1898), another one is The Mezzanine by Nicholson Baker (1988). Both were successful, and Baker's novel has spawned a series of similar works, but in popular genre fiction an antagonist, a conflict, and obstacles that lead to hardship of some kind for the protagonist are almost a must. Readers seem to experience a special kind of satisfaction when a character they identify with overcomes a difficult obstacle and achieves a longed-for goal. That kind of satisfaction is absent when the character just has everything they wish for from the beginning and keeps it until the end. But you can certainly experiment and maybe you'll begin a new literary trend. The current trend towards cozy fiction almost seems to point in that direction, emphasizing functioning relationships, positive emotions, and small, mundane, pleasant experiences over traumatizing tragedies and dysfunctional personalities.
Back in the 19th century, authors would sometimes write sketches - short works of prose, heavy on descriptions, that would introduce a setting and relationships between their characters, but contained very little if any plot development or action.
Lately, the genre of slice of life is gaining popularity - narratives that follow the everyday life of the characters, no big story arc.
There's also prose poetry, which may not even include any characters at all. Prose poetry doesn't count as stories, but that doesn't mean it's inferior or can't find an audience (albeit not a large one, it's fairly niche these days).
So yes, fiction without conflict is very much a thing, and has a long and respectable tradition.
One thing to consider, though, is that a conflict is a large part of what makes most fiction interesting. So without a conflict, you need something else to make the work interesting for the reader. Perhaps an intriguing setting, perhaps the unique characters, perhaps the lyrical descriptions or witty humour - but something.
You may also have noticed that works of this sort are usually pretty short, because there's no plot development that would change the situation, and too much of the same would get boring after a while. But if you know when to stop and start something new, you can keep the reader happy.
Kurt Vonnegut came up with the idea of graphing the happiness of the main character in a story: according to Vonnegut there are only 8 shapes for stories. There's a summary here, and an entertaining video of his idea here.
More important is what you do to keep your readers entertained. Here is an example of a successful story where nothing bad happened to the main character. I've forgotten the title and author (maybe Steinbeck?) of this novel (perhaps a collection of linked stories) where the main character had settled in a peaceful valley, after many years of struggle and poverty. Nothing bad happened to him thereafter, but it seems that his bad luck had somehow passed on the the inhabitants: somehow he managed to ruin their formerly idyllic lives, despite never trying.
I think what you mean are called multi-act dramas. Usually modern media uses a 3-act scheme, while apparently Shakespeare wrote mostly 5-act dramas.
Usually in the form of "comedies" and "tragedies". So either "Normal - Struggle - Happy Ending" or "Normal - Climax - Catastrophe". Particularly the hero's journey seem to be a popular trope. Cliffhangers are usually a feature of serialized entertainment, where you also utilize these act schemes but leave them incomplete either by not going all the way or by adding a new beginning as teaser.
That being said a story is technically just a series of related events or experiences. A plot already necessitates a chain of cause-and-effect points within a story.
Also you don't have to be as extreme as killing your protagonist's loved ones, shatter their hopes and dreams, destroy their existence and make some suffer unspeakable horrors. A hero's journey could also just be a literal journey. Leave your comfort zone, experience interesting stuff and end in your comfort zone again (maybe the old one maybe a new one).
Though technically leaving your comfort zone is still some sort of struggle or tension and you might want to add some shortcomings as motivation for that even if it's just "the milk was empty and so I embarked on my quest to the supermarket and you will not believe what I saw there". Which is technically a cliffhanger without someone hanging from a cliff and being in a situation of mortal fear.
It's not absolutely necessary to adhere to these formulas and structures for it to be a story. Though usually if people feel that they need to tell a story, it's usually to express something. So the picture of a journey from A to B or back to A' is often a natural fit, whether it's a literal walk or a metaphorical journey.
Having some "arc" (you could literally view the ascending-descending of tragedy or it's opposite for comedy as some arc) gives that some guidance, makes the story not as predictable as monotone movement in one direction and not as chaotic as tacking detours that don't serve the main plot. Same for smaller ups and down along the way if you want to make it more than 3 acts. Also similar to music, the change between repetition and novelty, tension and release, harmonic and dissonance often makes thing interesting and engaging, rather than predictable or overwhelming.
Not to mention that getting somewhere and arriving somewhere and having an arc, makes the story feel complete. So if you played with tension and release, you'd in the end leave your audience where you picked them up.
So even if you don't make it your goal to stick to these formulas you'd likely have a hard time avoiding setting the scene and establishing your world, introducing some sort of change and get things in motion, having things sort out and get back to equilibrium or a new equilibrium or having that postponed or given as task to the audience. Unless you literally just write a report, which is also a story, but even then having some arc makes it easier to memorize important plot driving events, which you wouldn't call as such in a non-entertainment context, but which could very well be seen as such.
The point of communication is to convey information and emotion. What information and emotion is your story trying to tell? Does suspense help reach that goal?
Genres that don't require conflict include utopia, romance, slice of life, parable, and children's stories (educational or slice of life).
Would my reader be disappointed?
This is the trickier part of the answer. It depends on how you present it and there's no single solution since some readers may have unexpected expectations.
Chekhov's Gun is where you introduce something that will foreshadow (and thus avoiding a deus ex machina) later use. If your story happens to have a gun on the wall or you mention that the neighboring country is very militaristic then the reader might expect conflict to arise.
Tv Tropes has a page named "The Law of Conservation of Detail" which explains that the more important something is the more detail it gets. If your story spends the first 10 pages detailing the relationship between Alice and Bob, then the reader will infer that the story's focus is lighthearted. If you then talk about Bob's gun collection but the entire scene is obviously for the purpose of character depth and more relationship building, then the reader won't think it strange when no one ends up getting shot.
What stories are usually expected to have is conflict. Conflict doesn't have to mean hardship or suffering for the protagonist - it just means some kind of problem they have to solve or obstacle they have to overcome. Maybe the problem is that the protagonist has to decide whether they want ice cream or apple pie for dinner.
Even that isn't necessary - other answers have given some very interesting examples of stories that don't have much in the way of conflict, so it's possible to have an entertaining story without any conflict, but I'd say it's definitely a lot more difficult to pull that off.
