Mac stuck after trying to install updates, startup disk missing

I tried to install some updates on my Macbook Air. I received a notification asking if I wanted to restart the Macbook to install the updates, I accepted and restarted. When booting, the Apple logo appears, the progress bar starts loading but then stops at some point showing the message:

A software update is required to use this startup disk. You can update now or select another startup disk.

I select the Update button, at which point the progress bar starts loading again, but it interrupts almost immediately showing the message:

An error occurred installing the update. Try again or select another startup disk.

I select the Try Again button, but the same message shows again. I then select Startup disk, and I am asked to Select the system you want to use to start up your computer. The list, however, is empty! At this point I don't know what to do, there are basically no options or buttons that I can press.

If I select the Startup disk button first, instead of the Update button, I get the same empty list of Startup disks to choose from.

Any idea what can have gone wrong and how to fix it?

The generic advice when a Mac can't start is to boot to recovery HD (internet, local or external as is most convenient for you) and run Disk Utility to check for connection, decide to make a backup of critical files if needed, then attempt a First Aid - repair pass.

• https://support.apple.com/en-us/HT204156

Assuming the First Aid works, you should be able to boot again or reinstall the OS from recovery. If not, you need to seek hardware repair, data recovery or try an erase install.

The reinstall instructions below get you to Disk Utility so you can look there and proceed.

• https://support.apple.com/en-us/HT204904