Yes, sticking a thermometer down there would be a great start.
Ideally, a root cellar should be close to 32F, but not drop below that for ideal vegetable storage. If I was storing canning, I wouldn't risk it that cold as it's one thing for a carrot to freeze and another thing for a glass jar to.
There are veggies - like onions - who like it a bit warmer, so ideal root cellars have colder zones and warmer zones.
You'll likely have more consistent temps on the north wall. My parents house had a walled off area that was insulated from the rest of the basement, but not from the foundation wall and it did the job unless we had a protracted cold snap. You want a quick project, but even recycled bubble wrap draped down to "trap" the cold of a north wall could get you a cool spot without it freezing. Ideally, you would check the temp first thing in the morning for a while to see what the trends are and how it tracks with outdoor temps.
Rats and mice will be attracted to the smell of
fruit and veg and they can chew through plastic, so the thicker the better or look for metal buckets/tubs. The tubs themselves need airflow, so consider using hardware cloth to make mouse-proof lids or drill small holes to allow airflow. That said, last year I used buckets with potatoes layered in sawdust inside a broken freezer inside an insulated out-building and they lasted far longer than any other system I've tried in the past. We generally get only a little below freezing weather.
I think it would definitely be possible to do this as a quick project. Friends I know who used to have a crawl space did get a pair of movers dolly's and fashioned a "cart" so they didn't actually have to wreck their knees crawling on
concrete.