Is there some way how to monitor free disk space from the Azure Portal?
I know that there are all kinds of diagnostics for I/O, memory, network, CPU, .NET, SQL, ASP.
This is not possible today through Azure Portal or Azure Monitor. Free disk space is a guest OS performance counter. If this is a Windows VM, you can use the Windows Azure Diagnostics (WAD) agent to collect performance counters to either Azure Storage table and/or EventHub and setup custom tool to monitor this data. If this is a Linux VM, there is also the equivalent Linux Diagnostic Extension.
Here are some relevant links on WAD -
https://docs.microsoft.com/en-us/azure/virtual-machines/virtual-machines-windows-extensions-diagnostics-template?toc=%2fazure%2fvirtual-machines%2fwindows%2ftoc.json https://docs.microsoft.com/en-us/azure/monitoring-and-diagnostics/azure-diagnostics-streaming-event-hubs