It's possible to generate a system that nearly "appears random." That's what Pseudo-Random-Number-Generators do. "Random" is more appropriately applied to a process, not the result of such process.
No written string of numbers is "random"; it merely is. The best one can do is say that a string of numbers obeys certain laws that randomly generated strings do.
Champernowne's number (.12345678910..., as you gave) does have the property that "any string occurs with the proper frequency." Thus (if mapped to a alphabet), it would contain the complete works of Shakespeare, the complete works of Shakespeare with one error, etc. (and the complete text of "Contact.") However, it can be proved (somewhere, I don't have access to review journals) that Champernowne's number does not obey the Law of the Iterated Logarithm. (I don't know how to generate a number that does except by ad hoc post hoc adjustments to the output.)
Thanks.
I'd never heard of the Law of the Iterated Logarithm; don't know much about random walk theory.