What are some decidable problems which cannot be solved in real life(due to time and memory constraints)?
Given an integer k, are there integers (positive or negative) x, y, z with an absolute value ≤ $10^{1000}$ such that $x^3 + y^3 + z^3 = k$? 

It is believed that without the limit on the absolute values the answer is "Yes" unless $k \equiv 4 \mod 9$ or $k \equiv 5 \mod 9$. On the other hand, solutions are so rare that most likely solutions for some values k cannot be found in the lifetime of the universe. 

And both the inputs, and the space needed for a computation, are tiny. The limitation on the size of x, y, z makes it decidable.