No, in fact we've proven there are a lot of things we can't do, and oceans of things that are theoretically possible but completely impractical to do (another class of people mistake intractable for impossible, but for small enough N sometimes it is both possible and worthwhile).
Undecidability and the halting problem is usually analysed for Turing machines (I am aware that there also exists literature on these problems for other types of automata). The problem is: It is impossible to build a Turing machine. Every computer that has been built up to know is just some kind of finite state machine. So the halting problem is perfectly (in a mathematical sense) deciable for real computer programs. The problem is: We have no idea how to do this in general in a timespan that is suitable for practical purposes.
Or my current favorite: a very senior and very optimistic person who apparently doesn’t believe in decidability, the halting problem, or NP completeness.