Trying to figure out if a modern computer is capable of manipulating arbitrarily large pointers
The universe is finite, so you can't build a real turing complete computer. 

The best you can do is build a computer which observes whether the program it is running is getting close to the limitations of the computer, and when it exceeds them, builds a new, bigger computer, with higher limits, transfers the code to the new computer, and starts all over again. 

PS. It is quite possible (and not really very difficult) to build a computer with 256 bit pointers. Since the universe is not just finite, but has a known size, 256 bit pointers are enough for anything.