Having actually owned and used a similar machine (Commodore PC-1, 4.77 MHz 8088, 640K of 150ns RAM), the performance of that website is impressive.
We're talking about a machine that was slow enough that typing fast on the keyboard could overflow the input buffer. You could actually follow the cursor drawing the screen in text mode (80x25), Xenon 2 ran on it at 2 fps. As for keeping it stable...
To be fair, if you ran only one thing at once on the machine, it could stay up for a long time - provided the drivers didn't cause an overflow, or a program didn't overwrite another's memory.