On the contrary, uncomputable functions are uncomputable relative to a given model of computation. Many functions cannot be computed by finite state machines but can be by stack machines, and many that cannot be computed by FSMs or stack machines can be computed by Turing machines. (The interesting fact here is that Turing machines are equivalent to a large number of other models of computation, and no "fair" model of computation is known which is more powerful than Turing machines or those other equivalent models. ("Fair" means that the model of computation does not do something that a mathematician with pencil and paper could not also do---"unfair" models do exist but they do things like assuming you can do an unbounded amount of work in a finite number of steps.))
The fact that the halting problem (or the nonperiodic tiling problem mentioned by someone else) is undecidable by Turing machines (and all of the other equivalent models of computation) provides an interesting philosophical tri-lemma:
1. There is a "fair" model of computation that is stronger than TMs. No such are known, and the evidence against them is that a bunch of other models turn out to be equivalent than TMs. There's a considerable amount of mathematical fame if you can find one, though.
2. TMs, etc., are the most powerful "fair" model of computation and there is no physically realizable "unfair" model[1,2]. This has some interesting implications for the "superintelligent AI" stuff. This is also the general consensus as far as I know.
3. Something out there is capable of performing "unfair" computations. (The Oracle of Delphi? Something woo-woo?) I think there are information theoretic/thermodynamic reasons for discarding this option, but what do I know?
[1] Note that a finite state machine is actually the most powerful known physically realizable form of computation. Your laptop is actually a FSM---it has a finite amount of memory and if you converted all of the mass in the universe into RAM it would still only have a finite amount of memory. All of the stronger models are mathematical abstractions.
There are computation models that are not obviously similar to computers; using DNA segments in a broth to solve instances of the Traveling Salesman problem, or soap bubbles (https://www.americanscientist.org/article/the-soap-film-an-a...). These are also equivalent to FSMs---there is only a finite amount of DNA or soap involved.
[2] The fact that functions exist (like the Halting Problem) that cannot be computed by TMs is not in itself terribly important beyond the implications of that tri-lemma above. Nor does the fact that essentially all functions are uncomputable by TMs (proof sketch: continuous numbers are transcendental with probability 1.0). I mean, the halting problem is a thing, so what? It only becomes important if something in the universe actually computes one of those functions that cannot be computed by TMs---that would rule out #2 and leave everyone scrambling for #1 or #3.