Is it possible to run a motherboard in distilled water?

I've done it. Don't do it.

I set up a computer in an acrylic case with good quality distilled water and a cheap motherboard as a test, with heatsinks only (no fans/moving parts). I cleaned the inside of the case with isopropyl alcohol, thinking that would remove any existing contaminants.

Within a day or two, I noticed that all the contacts/metal parts on the board began to rust. Even the stainless steel on the case of the SSD had begun to rust. Another day later, the motherboard died. When I removed the motherboard, being the first time anything physically removed (no fans), a huge cloud of rust particles came off and turned the water a lovely brown color.

Stick with something that metal parts can be friends with, like mineral oil.

Yes it is. Running a computer in distilled water is no issue.
However, keeping the water distilled is near impossible.

As soon as contaminants pollute the water even in very small amounts, the water will begin to corrode and given enough ionic contaminants, the water will stop being an insulator and become a very good conductor.

This kills the computer.

Now various people will say different things with regards to the amount of time it takes for the water to become contaminated enough to cause problems but in almost all cases it is within weeks in sealed environments, days in open.

Mineral oil is a far better alternative for a submerged build.

I would be highly surprised if it actually worked, even for a second. Motherboards have some pretty high frequencies, and the PCB routing is intricately designed to minimize capacitance so that they can actually carry these signals.

Changing the fluid that is around the board from air (dielectric constant = 1.00059) to water (80.4) is likely to introduce a lot of capacitances that weren't designed for and would be way out of tolerance, especially for channels like CPU to RAM. The additional capacitance just wouldn't allow the signal to switch fast enough to be able to reliably transmit the data. By the way, mineral oil has a dielectric constant of 2.1, so much less capacitancy than water, and some people have had success with submersion in that.

If you would be doing this so that you can overclock everything, then the higher dielectric constant works against that by reducing the maximum frequency that the board can operate at.

The Cray computers didn't have nearly the same challenges to being submerged, since the highest fundamental frequency signal on the board was 125MHz, and modern machines potentially have ~4000MHz signals, with common RAM being just below 2000MHz, with harmonics extending to >5x the fundamentals to form the waveform accurately.

I agree with the others here that have noted that metals are slightly soluble in water (especially copper), so the water would start to become conductive immediately. Voltage differences would also cause electrolysis through the water and H2 + O2 would be produced, as well as forcing ions into aqueous solution.