Hard boiled eggs are okay at room temperature for a couple hours, but beyond that, store boiled eggs in the fridge, unpeeled if possible.
If you prefer to peel your eggs all at once, store them submerged in cold water in the fridge. You’ll need to change the water every day, so it might not be much of a time savings. Alternatively, you can drape damp paper towels over the eggs instead of submerging in water, but it’s still recommended to swap them daily.