Asp.NET Core 2.1 HostedService - keep running on Azure
Ok, so in the App Settings in Azure, there is a setting Always On this worked for us. :)
We also found out from Azure support, that if you are on the lowest tier in their offered packages, this is treated as a "Development" environment and will only have limited up-time. As a result, we could expect the application to go offline when we reached that limit.
We argued that there should have been something more obvious in the dashboard for us to know this.
Once we upgraded to a Standard tier, the application stayed online.
Also, if you are running a hosted service and it hits an unhandled exception, this stops the service. You need to make sure you are handling exceptions for this to work.