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I assume you need to upload a shapefile to define the extents of data you're downloading.
If you want to make the shapefile smaller, let's look at the three parts basic shapefile parts and how you can make each smaller:
.shp: This stores the geometry of the shape. To make this file smaller, store less geometry objects. You could do this by deleting detail you do not need. This could mean deleting interior boundaries (islands/holes) when you only need the outside boundary. You could also reduce the number of vertices by generalizing the shape.
.shx: This file describes where the geometries in the
shp file start and stop. This usually isn't a big file so there's not much you can do to change its size.
.dbf: This stores the data attribute table. If you only need your shapefile for its geometry and don't need the attribute data, you can delete columns from this file.
There are other files that optional to shapefiles like
.sbx. They are not needed for the shapefile to work so they can be excluded from the uploaded files. You only need the three above.