I'd say there are two basic things you need to do.
First, make the story based on personal attributes, personal growth, relational issues, - Anything that is timeless. You can write about the insecurities of a teenager, wanting to find their way in the world. Or about a person facing their own mortality (perhaps they have learned they are terminally ill) and the personal journey they make in doing so. Or, about a psychosis. Romance. Greed, exploitation.
^ Whatever you choose as your story's theme, those are the things you try to focus on in your writing. Add to these as you can, to augment the strengths you find there.
Second, prune out anything that dates your story. You've identified slang and tech. You might also want to pay attention to the roles of people in society. Be aware if you are writing in a way that implies male dominance (suggests an older time) or gender equality (suggests a more recent time). Prune out anything that tips towards a time. Clothing, behaviors, cultural tip-offs like the role of religion - these things can date your piece. Make sure they are not doing so.
^ These are the things to remove from your story.
But, also, if you are strong on the human elements, on the realism of the journey that characters go through, on a compelling narrative about .... strength through adversity or some such, then in a sense you will have some leeway with the second part. Classic books like Catcher in the Rye or Moby Dick (etc) are 'timeless' even though they can be placed to a particular date. You want to avoid distracting with slang, tech, etc, but don't get hung up too much on that part of it.