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My dataset: a shapefile containing squares of 1km size. It's huge, this is just a sample: enter image description here

My aim: divide each cell so that the end result would be the same grid with 500m each cell. Something like this: enter image description here

Since the grid is huge I would need a way to make these squares smaller automatically. Any ideas on how to do it in ArcGIS 10.3?

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As user30184 pointed out, there's a tool that does the trick. Run the tool Create Fishnet Simply put your old 1km grid in the Template extent.

This will populate the coordinates of your extent.

Then fill the "Cell Size Width" and "Cell Size Height" with 500 geometry type > polygon

That's it

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If you want to take a Python script approach, try out this customer grid tool, http://www.arcgis.com/home/item.html?id=4e2a8fe3f297405d81747df1d1fdb45d. It should create quarters of whatever grid size you have.

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