I would grab copies of the files and database and put those on a development server running an old version of php.
Then upgrade your files and database incrementally through the various concrete versions until you get the 5.6.4.0
Next change the php version to 7 and see what breaks (fix it)
Finally replace your files and database on your live server with your upgraded ones.

Thanks so much for your reply! Alas, what you're suggesting is far outside my skillset. Being able to upgrade Concrete5 from within the dashboard was pretty much where my skill level ends (well, that and begging code to customize slide shows from good people here...)

My last-ditch hope is that I can at least rescue the site-content, but considering the integrated snarl that is the concrete5 'files' folder, that seems highly unlikely, yes?

(and before anyone scolds, yes i did back up the site to my own computer...the old one...years ago...that is long dead. argh.)

The vast majority of the 'site-content' is held in the database.
The files folder contains all your images ...
If you would like me to assist you with this site rescue, I would need to login to your server to grab copies of the files and database.

Contact me by private message if you would like me to assist...

Chances are your host updated the php version without you noticing. If you can get them to set it back to php5.6 at least temporarily, that may get your site up and running while @ConcreteOwl sorts it out behind the scenes.