Turing complete and computational power
If we think of the universe as finite, then anything that needs more memory than that finite amount is beyond our computational ability.

However this is not a good model for studying computability, the Turing machine model works much better in reality and that is the reason we use it for studying computation on real computers. A Turing machine doesn't really need an infinite amount of memory, it only needs an unbounded amount of memory. For example, we can provide additional memory to a computer over time (as the computer needs more and more memory) and then we have something similar to a Turing machine. If we assume that we have unbounded amount of time and memory for finishing our computation then the Turing machine captures this concept of computability in principle quite nicely.

Check the Wikipedia article about Turing machines, there is a section that discusses the relevance of the model.

If you are interested in feasible computability, then complexity theory (where we consider various amount of resources like time and space for performing a computational task) is closer to the what we can really do in practice than computability theory. Many experts state that the feasible computations fall in the complexity class $\mathsf{P}$ (and more recently in probabilistic and quantum versions of $\mathsf{P}$, i.e. $\mathsf{BPP}$ and $\mathsf{BQP}$).