I've come across a lot of users that have requested the ability to download ArcGIS Online Hosted Feature Services and ArcGIS Server Feature/Map Services. See that attached tool to do just this. The tool will also allow you to download attachments from ArcGIS Online Hosted Feature Services. Hope users find this helpful and useful.
Update: added Download Service 3.x.zip which utilizes Python 3.x and will work in ArcGIS Pro. The 3.x version will NOT work in ArcGIS Desktop. Use Download Service.zip for ArcGIS Desktop.