The way it works is that the entire assembly is in a vacuum. It kinda has to be as any gas which touches it will instantly condense to it or freeze to it. You then have a dual cryostat of liquid helium and liquid nitrogen cooling down the assembly (within the vacuum). The helium and nitrogen cryostat also have a vacuum shield. The nitrogen (liquid at 77K) is a sacraficial coolant which is far cheaper than liquid helium (liquid at 4K) that you need to get to these temperatures. Your're right that thermal radiation is an issue so you have to be careful with the placement of any windows or mirrors around the device.

Souce. I have a PhD in physics where I used equipment cooled to 4K.

Great, then we both have physics PhDs, and you'll know that none of that equipment has, or easily could be, sufficiently miniaturized, which is the topic of discussion ("extremely small cryocooler"). You can't put nested closed dewers of liquid nitrogen and helium on a O(1 mm^2) microchip, and the reason is exactly what I said: it will warm up too fast.

Ah, you're totally right. I misread the OP. Sorry.

No problem :)

What's wrong with attaching said microchip to a piece of copper for increased size? Genuinely curious.

To be useful in a data center you could cool a slab of copper the size of a fridge and surface mount thousands of chips on it.

The topic is cooling small objects so that personal electronics (e.g., your phone) can compete with datacenters. Cold at scale (i.e., in datacenters) is comparatively easy.

What kind of wavelengths are we looking at for the kind of temperatures used in this article?

About 700 micrometers, almost a millimeter. That's from applying Wien's displacement law to the boiling point of Helium.