I recently read to cover them with a thick layer of straw to protect the shoulders/exposed areas from freezing. I did this with carrots last year and it worked well. We have lots of grass clippings but I'm afraid they would mat down too much.

In mild climates you can leave them in the ground all winter and harvest as you need them. Where the ground freezes hard, a root-vegetable clamp is a good option. The following link will show you how to construct one. Scroll down to the article on clamps (page 17). Good luck.

Garden Journal - Clamps

Here is a link that might be useful: Alan Chadwick

either method is a call to arms for rodents of various types. They love the cover and they love the food. A root cellar lined with hardware cloth is more prudent.

32 is perfect, but 38 is acceptable too. You don't have to have a root cellar in Zone 8, just keep on growing root vegetables and pick as needed. Though I remember gardening in California two winters many years ago, got crops until the gophers found the garden. In your case, I would just have hardware cloth underneath, around to depth, and a hot wire around.

I have had an infestation of chipmunks for a couple years around the house (not in the garden though), they have started damaging brick paths so I am doing something. I have trapped five, but those I can not trap, I dig up their burrow, mix broken glass, and refill. They move on, usually only ten feet away, but at least not under the bricks. It is my fault of course, I have used spent wine grapes for composting once too many. They love having 50-100 lbs of seeds pre-buried.

Some root crops sweeten with frost. As far as I can remember, beets don't. I would harvest them and either can them or keep them in moist sand at cool temps.

Glib, I've got the hardware cloth and raised beds. Also new raised containers, no bunnies or squirrels (maybe turkeys, not sure!)
I grew beets a couple of years ago, got distracted, and when I went to check they had gotten all woody!
I don't can any more (might start again since I've retired), but can they be frozen? I have a Food saver and freeze most things I grow/sauce/chop/etc. Nancy

I don't know if beets freeze. But I think that in Zone 8 they can stay in the ground uncovered, unless your rodent pressure is overwhelming.