How do you play minecraft without the scary monsters?

There's one way to do it without mods.

While in your game world go to your settings and turn the game to "Peaceful". This will prevent any hostile mobs from spawning in.

You can also play in Creative Mode. In that mode monsters will appear, but they will not attack you. Of course there are other differences in Creative Mode as well, so you might not want to use that one. You can stack Creative and Peaceful if you really want both.

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If it's just the skeletons that bother you, you can mod Minecraft to exclude them. In Control or Mob Spawn Controls 2 are each one of several mods that let you do this.

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Edit: I usually play without monsters, as I prefer exploring the world and building far more than I enjoy hunting monsters and hoping they drop material I need.

Admittedly, I also play heavily modded Minecraft, as I think many mods add interesting things to the games.

By eliminating them entirely you do yourself a disservice.
The Minecraft experience is not just about building, but about braving your surroundings and earning your materials, so you actually feel a sense of accomplishment.
Build a 2-block high wall so they can't reach you. Set torches so they don't spawn near you. Use a bed to skip the nighttime so the sun can smite the undead. There are ways to play defensively, but you should learn to face your fear.

Play on peaceful mode