Hey guys! EtreCheck is going to be hard to run because the computer won't boot.
"Can I bring it into the Genius Bar to get it cleaned up?"
You are asking end users here, not Apple. Different Apple Store Genius bars likely have capabilities.
History here suggests that your computer (21.5-inch; 2013), if equipped at the entry level, has a very slow and under-spec hard drive that gets slower with age and abuse. Your symptoms sound like the hard drive failed requiring full disassembly, and that is something the Genius bar would not likely attempt at the counter.
Also, per their published service documents, they may not work on a computer that is classed as "vintage" or "obsolete." The Late 2013 iMac is classed as "obsolete." They may evaluate it but not fix it.
👉🏻 Before doing anything, try booting into Safe Mode:
How to use safe mode on your Mac
That bypasses unneeded software that may cause issues and also does some disk/file repair. If you get a useable desktop in Safe Mode, STOP! Let the computer idle in safe mode for 1-2 hours to give time for all the safe-mode routines to finish.
You can stay in Safe Mode to off-load files and gain space.
Once you have done that, see if the computer starts with a normal boot.
If you cannot get into Safe Mode, then the drive could be truly dead and you will need professional help replacing it. Because the case was not designed to be opened, and the the 21.5-inch iMac that year was a bear to work on, expect to pay an independent tech US$60-80/hour to service it; figure two hours of labor on top of the cost of a new drive and the required reseal kit.