Indeed. Getting the inputs to be exact is "challenging" for an analogue computer too ;) And there are issues with noise.

You could probably build a computer to do it if there are uncountably many parallel universes to exploit when you build it, I suppose.

I was going to mention an Oracle Machine being able to do stuff like that but then I saw this on wikipedia:

Oracles and halting problems

It is possible to posit the existence of an oracle which computes a non-computable function, such as the answer to the halting problem or some equivalent. A machine with an oracle of this sort is a hypercomputer.

Interestingly, the halting paradox still applies to such machines; although they determine whether particular Turing machines will halt on particular inputs, they cannot determine, in general, if machines equivalent to themselves will halt. This fact creates a hierarchy of machines, called the arithmetical hierarchy, each with a more powerful halting oracle and an even harder halting problem.

http://en.wikipedia.org/wiki/Oracle_machine

One more comment on the computable numbers that Paradigm Shifter was referring to: Using an algorithm to determine the n-th digit as the representation of a number might not be all that useful, because many simple things you may want to do (e.g., check if it is larger than 0) are not computable.