If you are the administrator of the service, you will probably have access to the data. In that case you could use something like the ArcGIS Open Data Help | ArcGIS which has feature to allow users to download the data.
In case you are not the owner of the data, you can use the REST API to query the data. A simple way to do this is using some Python code. A small example can be found here:
https://community.esri.com/message/436087#436158
Please note the services by default are configured to return no more than 1000 features. If you are not the administrator of the service, you will have to do multiple requests on the REST service to obtain all features. You can provide the REST query with a geometry to restrict the results to that geometry.
Kind regards, Xander