It got closer and closer, and there was nothing they could do to stop it.
If you don't want gory horror, then fundamentally horror is about losing control and a sense of violation. For intellectual horror to work, you strip away people's ability to deal with the things in their lives they must deal with.
A horror story could be as simple as an old woman who has fallen in the kitchen and can't reach the phone. As long as you can keep adding levels of suspense and anxiety to the situation, ratcheting up the levels, it keeps getting more horrific. The old woman has a bit of food and water. Then these are gone. She has no bathroom and soils herself. She dwells on how she will be taken from her beloved home and put in a nursing home. She's angry with herself for her failure to get an emergency button. The injury she suffered falling gets worse. Mice become threats as she feels them scuttle over her when she's weak and asleep. Her son was supposed to put out traps, but didn't. Finally, they start biting her and there's nothing she can do to stop it. She's weaker and weaker, more and more hurt. Finally, her nephew arrives (who she trusts and has put in her will), and rather than help her, he leaves so she can die and he can inherit her house. Then the mice come back out, staring at her.
Horror takes our lack of control over things in our lives, the things we feel helpless in the face of, and our deep-seated fears, then it keeps adding new levels of badness. The best horror story I read (Survivor Type by Stephen King) was a short story where a former surgeon is the sole survivor of a shipwreck, stuck on a barren rock in the ocean with a kilo of heroin he was smuggling. He injures himself trying to catch a bird to eat, amputates his foot, then eats the foot in desperation. Finally he starts amputating more and more of himself to stay alive.
The man started out a dishonored surgeon, desperate enough to be smuggling heroin, then placed in a shipwreck, abandoned on a rock, starving, then crippled, and finally forced to slowly eat himself. All in the course of (I think) 15-20 pages.
At each step, you can sympathize with the person, understand their pain, and move on with the story. Barring supernatural horror, each step is possible but unfortunate. But the levels keep needing to be amped up with each step. The violations can be physical, mental, emotional, or spiritual. Ideally, they are all of the above. And unlike a traditional story, you don't get to solve your problems. If you do solve a problem, the solution is it's own terror OR is replaced by two new ones as soon as it's fixed. Then it's up to you if you want a happy ending. But the standards of happy are pretty low in a horror story.