Proof Idea: There are irrational numbers whose decimal expansion cannot be computed
You can't even write a program that outputs one single irrational number. What you can do, is write a program which will output for example every decimal digit of some irrational number, one after the other, assuming unlimited resources and unlimited time, and which will output every single digit eventually (but will never output all digits). We count that as "outputting a number". 

If you tried to write a program that first outputs all digits of π, followed by all digits of e, that wouldn't work because it never will finish outputting the digits of π. What you could do is output the first decimal of π, followed by the first decimal of e, followed by the second digits of π and e, and so on. So a program can output two (or three or four or a billion) irrational numbers, if we change the definition a bit. 

You could write a program that outputs a countable infinite number of irrational numbers. Start with the first digit of the first number. Then the second digit of the first, followed by the first digit of the second number. Then the 3rd digit of the first, 2nd digit of the second, and 1st digit of the third number, and so on forever. So this program would eventually output each digit of a countable infinite number of irrational numbers. 

You can't extend this to an uncountable numbers. Being uncountable, it is impossible for example to output just the first digits of all these numbers. 

So the best you can do, with a modified definition of what "outputting an irrational number" means, is that you can have a countable number of programs, which each calculate a countable number of irrational numbers, but countable times countable is still countable, so you cannot output all irrational numbers.