> Can't you say that any single (shortish) number is random? How you can demonstrate 99999999999999999999 isn't random?

You can not. Randomness is a property of the procedure used to generate it, not the number.

Just looking at it with your eyes you can't. But there are statistical analysis techniques used to find bias in random numbers, patterns in random numbers generated using the same RNG and this is often used to demonstrate that some technique or generator is broken and cryptographically insecure. If a 128 bit number doesn't provide 128 bits of security then something in the generation of that number is broken.

There are other comments in this thread that get into some nitty gritty details of this that I don't pretend to be an expert on.

You're basically applying the Sorites paradox to random numbers.

Sure, asking whether or not any single number is random does not really make sense (although I suppose if we have enough digits to look at then something similar to Benford's law might apply to their distribution). But the point being made was about humans as random number generators and the distribution of numbers they produce. It's like comparing individual molecules to statistical physics.