How are multi-month exposure photos possible without the image turning completely white?

(There are people here who know a lot more about this than I do, but since there is no reply yet I'll give it a shot.)

There are two questions here: How did Michael Wesely do it? And could we do the same with a digital camera?

Digital camera

I'll start with the calculations for a digital camera, it's the simplest:

From an exposure table we can pick a level of lighting (exposure value at ISO 100) and read off the corresponding camera settings.
I'll assume that average daylight is EV 14 at ISO 100, which wikipedia describes as "typical scene in hazy sunlight (soft shadows)".

The lowest ISO and smallest aperture available on my DSLR with a kit lens is ISO 100 and f/22, which gives an exposure time of 1/30s for the "hazy sunlight".

If we want longer than that, we'll have to use ND filters, dark pieces of glass/plastic that block a significant fraction of the light. (Think sunglasses or welding glass.) ND filters are specified in stops, one stop blocks half the light and doubles the exposure time.
The image doesn't turn white because we take care to block just as much light as we need to get the exposure time as long as we want.

The strongest ND filters I know of are 10 stops. But we could buy several and stack them, so our exposure times could look something like this, with a doubling for each stop:

• original: 1/30 second
• 10 stop ND: 30 seconds
• 20 stop ND: 9 hours
• 30 stop ND: 1 year

Once we get past a full sunrise-to-sunset period, we need to take into account that the sun doesn't shine 24 hours a day. For simplicity, let's assume that the sun shines 12 hours a day and there's no light at all at night: Then we need to double the exposure time again, since the photo is effectively only exposed during daylight hours, and we end up with about 2 years of exposure time for 30 stops worth of ND.

In principle, we can get the exposure time as long as we wish by stacking enough ND filters.

In practice there could be issues with vignetting (get larger filters), color casts (convert to B&W, or correct in post), and power (AC adapter and uninterruptible power supply). As well as other potential issues: Sensor noise? Do I really want to wait another two years for the long-exposure noise reduction? And can a digital camera even do a two-year exposure without breaking?
But in principle, it's easy.

With a digital camera we could instead take a series of photos with shorter exposure times and merge them in software (averaging). The effect is pretty much the same, as long as the time between consecutive exposures is trivial compared to the time spent on each exposure.

Michael Wesely

He's using a 4x5" view camera and a pinhole lens.

Pinhole lenses have very small apertures, perhaps f/256 to f/512 for a large format camera.

He would probably use the lowest ISO film, photographic plate or photographic paper he could get hold of. Certainly lower than ISO 100.

Film also needs to deal with something called reciprocity failure, where the specified ISO value is only valid within a narrow range of exposure times, e.g. 1/1000s to 1 second. For exposures longer than say 1 second, the ISO drops exponentially as the exposure time increases.
The exact effect depends on a lot of factors, so it's simplest to determine exposure time through experimentation.

From the article linked in the question:

It took Michael months and months of experimenting to make sure the negatives weren't going to be over-exposed. He said, if you'd planned to expose for a year you would have to do an exposure of 6 months, and 3 months beforehand and so on. You would have to collect a lot of data and find solutions for a lot of detail problems.

So I can't really say how many stops of ND filter he would need - more than none, a lot less than for digital.

But in general, that's how it's done:

• Low ISO
• Small aperture
• ND filters
• Experiment to figure out how reality differs from the calculations.