Hello Abby, and welcome to the Forums.
This cactus is in bad shape. I think I see some green tissue beneath the spines. If so, then there is still hope, but not much.
If this cactus is going to be saved here are some things to try.
First, re-pot it it in a fast-draining succulent soil mix. This should be one with more mineral content (pumice, scoria, perlite, or coarse sand). The container should also allow for fast drainage. Does the current container have drainage holes? If not, switch it out.
This looks like this might be one of those pots that have the saucer glued or glazed to the pot. Those are bad for any plant that requires good drainage. Even if the saucer is removable, do not leave it flooded with water and don't use it at al if it seems to impede drainage.
Second place this cactus in bright light. I would suggest putting it outside if this was Spring or Summer but in Colorado in October, lets keep it indoors. A sunny window with southern exposure would be best, but western or eastern exposure might do. Air-conditioning should not be concern.
A healthy cactus could/should be allowed to rest over Winter in this bright location with little to no water and no fertilizer. However, this is not a healthy cactus. It needs to recover immediately, another six months of not-so-benign neglect* will not do. We are going to try to give it the Summer id didn't get in Winter. In its warm, bright indoor environment, this cactus will now begin to be watered regularly. Water and then wait until the soil is mostly dry before watering again. It does not need to bone dry, just mostly dry. Of course the drainage must be excellent, as I already mentioned. Don't let it sit in a wet saucer after watering. We are trying to reestablish roots and get this cactus to take in water.
Beginning with the second watering, and continuing monthly going forwards you will fertilize this cactus with a complete liquid fertilizer which can be applied during water. Standard all-purpose fertilizer is fine. No need to buy a special novelty cactus fertilizer.
All I can add to this is good luck and remember that caring for cactus is skill that must be learned with experience. If this cactus does succumb, start again in Spring with another healthy specimen, and again try to apply the culture that is discussed here.
By the way, I'm not certain which cactus you have. At this stage it is quite hard to identify. 'Silver Ball Cactus' is a name that could be applied to more than one cactus. However, this isn't Parodia scopa, which is one species commonly referred to as 'Silver Ball Cactus'. Fortunately, the care recommended will work for any cactus.
* If there is any take-home message for readers, it is that the adage "Cactus thrive on neglect" is false. Cultivated, container-grown cacti need water, fertilizer, and regard as much as any other plant.
