I need to build a polygon (5x5 km) centered on a coordinate X,Y. I want to use this 5x5 square for an eventual clip.
How should I proceed in ArcMap?
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I'll assume you have your center points and their XY coordinates in a table. Step 3 requires an ArcGIS Advanced license.
Create a point layer using your coordinates as the points. You can do this several ways, one is using the Add XY Data tool in ArcGIS. You just point the tool at your table of points. You'll need to specify which field is the X coordinate, which is the Y, and the coordinate system the points are in.
Buffer those points by 2.5 km, set the Dissolve type to NONE. That will give you a layer of circle features centered on your points, each with a diameter of 5km.
Use the Feature Envelope to Polygon tool, using the circle layer from Step 2 as the input. That will convert the circles to squares, 5km on a side.