What happens when the startup disc of a Mac is COMPLETELY full?

Thanks so much to both of you, for all of the help. I was hoping that wasn’t the case. I guess the diagnosis is clearly final, but I have a couple of additional questions if you would oblige.

(N.B.: I have no backup of the system and why I lack said backup is a whole 'nother saga of Murphy’s law and drive failures that’s almost comical enough to share here, but not worth it)

This is a mid-2011 Mac Mini. Intel Core i7 2.7 GHz w 4MB Ram and 2 Hard Drives - the aforementioned problem child is a 250GB SSD, and there’s a 750GB HDD. The one bit of good news is that I used the SSD for the OS and applications, but kept our user files and all of our “data” on the HDD - so I was able to copy all of our photos, docs, etc. to an external drive and I can restore those.
1 - I assume I can just reinstall the OS on, and run on the 750 GB drive as a stop gap until I figure out my next move - is it that simple, or do I have to do anything else to make the HDD a “primary” drive?
2 - If the SSD drive is that fouled up, and stuck in this read-only status, how can I get files from it with an external case? (I have a SATA-to-USB adaptor somewhere and I might try it just in case there’s anything worth salvaging. This would be a just-in-case measure because-as mentioned-this drive mainly had OS and apps that can be replaced, all the important data was on the 750GB HDD)
3 - Is it worth investing in a new SSD to keep the old dog running IYO?
Pros: Only cost ~$150 (plus I might as well drop another ~$100 on boosting the RAM while I’m at it)
Cons: Risk of my sausage fingers fouling something up when I go under the hood, risk of some other HW component crapping out soon
Buy new mini:
Pros: latest and greatest specs, no sausage fingers under the hood
Cons: Costs $700-$1000

Once again - my sincere gratitude for all of the help you both provided.

PS - any SSD drive recommendations if I do replace the SSD?

Yes, you should just be able to do an OS install on the secondary drive, and then choose it as your Startup Disk.

If the file system is Mountable, you should be able to put the SSD in an external enclosure, and it will mount as read-only, and then you can do a Finder drag-copy, or use something like CopyCat X to copy the files.

SSDs are getting pretty cheap these days - if you still have use for the Mini, plop a new 500GB SSD into it for < $150.

Replacing a hard drive in that era Mini is not too difficult, just tedious.

There is nothing wrong with a Mini that old - I’m actually using a very similar one. Absolutely increase the memory. 4GB is a bit squeezy. (Indeed it isn’t impossible that the low memory has contributed to your SSD’s demise by forcing the system to page so much that you have worn the SSD out. Although a 250GB device one would really hope was large enough to cope with the page traffic, there could have been some pathological paging behaviour.)