you could cut off or clean out the rotted areas/sections and give a dusting of antifungal powder... Or, try some cinnamon powder... yes, like on doughnuts or mixed with sugar. I've been told it's a good antifungal. I've tried it only once and it seemed to work. Then let it dry out for a few weeks. You could probably leave if out of the ground for a few MONTHS and it would be fine. But let it get good and dried out on the cut sections and then repot the plant. Anyway, here's the images incase someone else would like a look and add some advice.

Ok here's an additional photo of the barrel cactus in the process of repotting. It isn't the best quality photo but gives an accurate picture of what color each part of the cactus is. The brown color is not dirt; it is the actual color of the cactus. Please let me know if you think this is normal or if it is rotted. And if it is rotted, if you think it can be saved, and how. Thanks!!

A picture would be helpful, but if it is rot you can save it by cutting off the rot, letting the wound dry and rerooting the remaining plant. It will take a while, but if you don't do anything it will spread and kill the whole plant. Or you can just root some of the pads. I just lost a plant that got the same treatment as all of my other 100+ plants. Sometimes it just happens, but in my case I think it was root mealies. If I would have taken the time to unpot it and look at it, it would be in better shape today. I had to cut 3/4s of it off and now I'm hoping for the best. When I finally took it out of the pot, it did not have one root. Sometimes there can be a slight cut or nick and the plant will get rot that way as well, so it's hard to say. Here's a link talking about rot: Causes of rot And here is a link to a thread that explains what to do in more detail Can my cactus be saved? Good luck with whatever you decide to do. It always hard to lose a plant.

Rachel, it's very hard for me to make an assessment when I am not physically there to examine your plant. I don't quite understand what you mean by soft but not overly. I see that your last picture you had lost a couple of stems, so perhaps if its really rot setting in the base, you need to catch where is the rot happening at. If it's right under all the stems towards the roots, you are better of cutting off as much rot/brown portion as possible until you see healthy green stems, and try to root the cutting as best as you can. At this point, you can try to snap 1 or 2 healthy cuttings at the tip end towards the base. Always let the cuttings dry up and callous before their to root them in new soil. And also make sure when you snap them apart, the stem must be close as to the root base as possible, if you can snap some roots with the stem even better.