We host our own nuget server in azurewebsites. To make backups of the packages we use Kudu console.
Kudu is the engine behind git deployments in Azure Web Sites. It can also run outside of Azure.
Read more info about the Kudu console.
In general open a browser with [yourwebsite].scm.azurewebsites.net. After login the Kudu interface is shown. I’ll be using the Debug console from the top menu.
With the file explorer you can browse your websites folders and files. Downloads are offered per file or per folder (as a ZIP). The Console window displayed below the file explorer is ‘in-sync’ and offers cmd or powershell access.
Be careful not to break anything 😉