Any given computation either halts or it doesn't. You can encode that information in a single bit, as a specific number. Since there is a countably infinite number of possible computations, you'd need a countably infinite number of bits.

So you can never find enough storage to hold the full solution of the halting problem in the real world. But you can find enough storage in a real number. Because real numbers can have a countably infinite number of digits after the decimal point. So you can stuff your countably infinite number of bits representing the solution of the halting problem in there.

Which specific real number you get depends on the details of the encoding, but it's definitely some real number. And it cannot be computed, because if it could, you could read the solution to the halting problem off its digits, but the halting problem is known to be uncomputable.