This sort of depends on whether you're talking about "randomly selecting a real number" in the real word or in the mathematically abstract world. In the real world, you can't do it, because almost all the real numbers are non-terminating decimals. Or to put it another way, any selection process is a computation, and almost all the real numbers are uncomputable and so can't be found by any computation.

To randomly generate a real number in the abstract, I suppose you could do a random walk of infinite steps on the real number line, with each successive step randomly going left or right and shrinking the distance walked. Off the top of my head I'm not sure if that's rigorous, but it sounds like it would work.

Simply write its binary expansion, by flipping coins, (heads=1, tails=0), forever.

Very carefully.