John,
If you are using Closest Facility, make sure that you put a cut off (say 10 miles) and find 5 closest stores AND make sure hierarchy is OFF. It should solve without running out of memory. However I will suggest that you use OD Cost Matrix solver and not the Closest Facility as it uses less memory (it does not need to return the true shortest path geometry) but otherwise works similarly. Just do the same settings, 10 mile cutoff, find 5 closest and Hierarchy OFF. You will be loading your 10000 customers as Origins and the 1000 stores as destinations.
I did not understand your second comment:
"I need to get to the point to where I can measure that all customers are 1 mile away their closest store, 2 miles from the 2nd closest store, 2.5 miles from 3 closest store, etc. You get the picture."
i.e. what is that you are trying to achieve? Are you trying to open more stores such that all customers are within 1 mile?
Jay Sandhu