How important do you think the vacuum really is? Wouldn't it make sense for a DIY unit to just cool water to 10C and then to submerge your hands or feat in that? That would be a much more efficient method of heat transfer.

The vacuum is the key to the whole thing. The blood vessels in your extremities are very good at contracting when they get cold, to help preserve your core temperature. The vacuum draws them back to the surface, meaning you can cool much more blood more quickly.

I made a half-arsed attempt at replicating this a while back. The stuff I noted for my next attempt:

As your vacuum increases, the hand/wrist will be pulled further into the chamber. You have to have something to hold inside the chamber that is anchored to the chamber, or a container where having the fist jammed up against the back wall is reasonably comfortable. (My original attempt: using juice concentrate cans rather than flowing water through a handle. I wanted to avoid flowing water if at all possible.)

Due to the whole "arm getting pulled inside" thing, the wrist seal is more troublesome than I hoped. My next try is going to use a dry diving suit wrist-seal.

You want as small an interior volume as possible if using a manual air pump. It's much more time consuming than you'd think to hand-pump the air out and achieve the desired vacuum. I wouldn't want to risk using an automatic air pump in this case, myself, so I'm sticking with the hand pump while reducing the interior space of the container.

Let me know if you have success with this; or, heck, even if you don't, because you'll probably have some good advice as a result.