How can I tell whether beetroot is still edible?

Generally, raw beetroot shouldn't become soft. The thing is that there are differing definitions of soft. If it is soft like a tomato is, then it has probably spoiled. If, on the other hand, it is still firm but has little "give" to it, then it's probably alright to eat.

It is the cooking that softens it up, though it can be eaten raw. I grate some raw beetroot into salads and stuff occasionaly, just for fun.

I'm not sure what to say about the smell thing. I think that raw beetroot generally smells a little earthy, especially if you haven't peeled it yet. The very distinct beetroot smell is usually released by cooking.

Obviously this is all a bit vague, as we can't send smells over the net yet.

Root vegetables are known for staying edible many months when stored in the right conditions: cool and humid. Traditionally this was a root cellar under the house that stayed above freezing in winter, yet cool in summer. Your refrigerator veggie drawer is made to do the same thing. The veggies you see in supermarket have been kept that way for transport and storage (sometimes even covered with waxy coating to preserve moisture). When they're brought out for display in the produce section, they are still kept cool and misted periodically to stay fresh-looking. If you don't keep them in those conditions they will become limp or soft, which can sometimes be reversed by returning to a high humidity environment.

All that to say, it's normal and probably reversible, but don't worry, people have been eating and storing beets since long before refrigeration.

I can say that a cut to stripes beet that was in the fridge for a few days became very dry and very bitter, I ate some and got bad feeling in my throat that still lasts after few hours. I recommend that if it's bitter trash it.