a root cellar would be great, but not everyone can have one (water table too high, etc). My FIL has a huge garden and orchard, and does not sell anything. They store, give away, or process everything. For this purpose, he built a walk-in cold room in his garage. He altered an A/C window unit thermostat so that he could get it to cool down lower than the pre-set temp limits. But it is not refrigeration temps, so no meat or dairy storage, for sure. It has worked well for him for at least 10 years. He lives in a very dry climate, and I am not aware that he adds any humidity, but I'm not certain on that. We've talked about doing this in our barn as we have an area under the loft stairs that would be a workable and convenient space.
However, for now, pop in your contacts to help resist the crying (always helped me, but alas, I no longer use contacts) and borrow a friends food processor, if you are able, and slice or chop away! Out side, perhaps.
Congrats on the large crop. Ours wasn't great - many rotted in the ground or didn't grow very large. But, first time ever growing onions, so I'm on a learning curve.