I'd say it's perfectly possible for a Turing machine + a source of randomness to generate a random real number in [0, 1). We just lazily generate more and more

I'd say it's perfectly possible for a Turing machine + a source of randomness to generate a random real number in [0, 1). We just lazily generate more and more digits to the required precision. It's no different from, say, pi being computable. We can even prove that there's zero probability that a rational number is generated: the decimal representation of a rational always has a repeating group of digits, but since each digit is generated randomly with 100% probability there will be no periodic pattern.