We have a root cellar. I keep the potatoes in burlap bags (garage sale, but they are easy to find at ranch supply places) inside of laundry baskets (dollar store ones). This keeps them dark and they have air flow. I tape labels to the basket and they are separated by color and size (we grow our own so ymmv if you buy instead). It gets warmer in there around late April and by June the potatoes are sprouting, so I go through and remove all sprouts or it can become a tangled mess. Potatoes will get a bit soft and wrinkled but they are still safe to eat. Just have to peel with a knife at that point. I've never had a rotten potato with this method.

It's true you shouldn't store onions, apples, and potatoes in the same place, but our cellar is small and I do and it's never been an issue. But I believe it is bc of the cool temps. I take out the onions when it gets warmer.

I store both onions and apples in laundry baskets. Loose, no bag.

I store beets in buckets with holes drilled in the bottom. They are packed in peat that I moisten before stacking them in there, layered pretty tightly. I water the buckets once a month and they have a loose piece of plastic on top (not a lid, just a piece of random plastic) to help keep the top inches from drying out. Beets will last about 11 months this way. They often grow some little leaves, nothing to worry about.

I have stored carrots just like the beets and they are okay but no matter what medium I use (woos shavings, sawdust, peat, soil) they always seem to pick up the flavor of the medium and I don't like it. They are better if I store in a spare fridge, loosely in a plastic bag with a little (tablespoon only) water. Roll everything around so each carrot has a layer of water, dump any excess. We usually have about 100 pounds of carrots and this has worked the best but if your cellar is cold all year that might be best for you.

Number one thing is to have a temp monitor. I have a digital one that we can easily read, there's a sensor in the cellar and the base is outside where I can see it often. You can set an alarm to sound if it gets close to freezing temperatures if you want. Our cellar never has gotten below 34 fahrenheit but it helps me know when things are getting too warm in the spring.