Startup can't get past circle with a line through it

I noticed my iMac was running rather choppily so I rebooted. I got a screen with the "prohibitory symbol" on it and followed all the protocols of the article If your Mac starts up to a circle with a line through it - Apple Support. Nothing works and the result is always the same - circle with a line through it. I can't get to Recovery Mode, nor can I start in Safe Mode.

Any suggestions as to what to try next? Thanks!