What's going on here? Leaving started getting covered in what looks like white mold out of nowhere.

lol I think you named greg problem in your question. White powdery mold.

Looks like a pretty bad case of powdery mildew, a fungus that spreads via spores that is extremely difficult to get rid of and can very easily and quickly spread through a collection.

The most crucial step to take right now is to isolate this plant away from any others you have, preferably in a completely different room. To begin treating the plant, use a damp cloth to wipe away the spores on the leaves. Toss this cloth in a ziploc bag and dispose of it in a closed trashcan. I'd typically recommend removing affected leaves, but it seems that the vast majority of the leaves on this plant have powdery mildew. So, you're probably just going to want to move directly to treatment. Unfortunately, there is not a one size fits all treatment for powdery mildew, meaning a treatment that works for one person may not work for you. The most commonly recommended treatment is a copper fungicide. Diluted hydrogren peroxide, diluted milk, a combo of rubbing alcohol and fungicide, and potassium bicarbonate, among others. I'd recommend choosing one treatment, trying it for a bit, and if it doesn't seem to be working, moving on to something else.

I wrote up a post awhile back about powdery mildew detailing what precautions to take and the various treatment methods. It is available here and also linked in the FAQ.

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Make sure when you water the plant, you water the roots, not the leaves. I use a tiny piece of paper towel to dab up any water drops on the leaves after I water them. Is the humidity high? That could also be a factor. If it is fungal, you can mist or gently rub (I use a small, soft paintbrush) the leaves with 70% rubbing alcohol once a week or so until it clears up.