If it is your server, I assume that you have access to the server (the folder structure which is part of the ArcGIS Server installation). If the data was published to ArcGIS Server with the option to upload the data, it will be part of the service directory and it will be faster to copy the data from there instead of doing multiple REST requests. If for some reason you need to download the data using REST requests, you can also have a look at the tool developed byJake Skinner and available here: https://community.esri.com/docs/DOC-6496-download-arcgis-online-feature-service-or-arcgis-server-featuremap-service
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