"Have I accidentally deleted the operating system?"
-----------
Not necessarily. If when you start the Mac normally you get a login screen, the OS is present. If you get an error screen, you need to show that.
What the message is saying is just what you see, an error has occurred and the system can't proceed. Why? No idea as many things can cause that.
If it were my Mac, I'd cut to the chase, wipe the drive and install the OS. I'd then either use one of my several backups to restore data or maybe just start from scratch again. But that's just me. You may not have backups and/or just don't want to chance a disk wipe.
The Mac is eleven years old and you likely have no idea of any problems it was having before you got it. It's a personal decision as to how much effort you want to put in to resurrect and old obsolete device. If you have no backups and want the data on it, I can certainly understand trying to "get it back".
I suggest you wait for others to respond with other ideas.