The "over 300 billion years" kind of headlines always bother the crap out of me. The clock won't run for 300 billion years. It'll probably run for ten or twenty.

But during that time there will be several copies of it built, and they'll run in several locations around the world, and the outputs of those copies will deviate less than some tinysecond per longtime. Attoseconds per month or whatever.

And the useful timescale is more like, how much do they deviate over five seconds?

Because they'll be used for something like astronomical ranging. Say you have two radio telescopes at distant points on the earth, both observing a distant stellar source simultaneously. The long baseline forms one side of a triangle, and the distances from the two observatories to the star form the other two sides. The more precisely you can determine those distances, the better you can resolve the position of the source. So you've got coherent receivers that can measure individual wavefronts in the signal, but how coherent are they, anyway? That depends on the clock they share. But they're on opposite sides of the planet, they don't actually share a clock...

And the measurement period over which you can compensate for other confounding factors (probe and measure the atmospheric distortions in the way, for instance) might be very brief, so the clocks' stability over brief periods is salient.

This is my lay understanding, so if someone more versed in the art wants to chime in, I'm all ears!

"One second per 300 billion years" is really a statement about the bandwidth of the signal that the clock generates, translated for lay readers. We don't know what the actual bandwidth of the signal is, since it's not stated in the article. Bandwidth is measured in Hz.

In order to measure the bandwidth of the signal (and how accurate the clock is), you have to measure the clock over a period of time. The shorter that period of time, the worse your measurement will be. The smaller the bandwidth, the more time you need to measure it.

Note that a signal with 5 Hz bandwidth and 5 Hz center frequency would make a very terrible clock, while a signal with 5 Hz bandwidth and 5 GHz center frequency would make a very nice clock. The bandwidth is the same in both cases, and the amount of time required to measure bandwidth is the same, but they make very different clocks.