17 753 889 197 660 635 632
interesting, that is rounded 18E+18
One number 0..35 takes minimal 6 bits. You can easily encode a 6x6 magic square as a 5x5 square so that you need to solve only the last square per row/column. That means you need 25 x 6 bits = 150 bits ~ 19 bytes (almost a factor 2 over the 36 bytes normally) There might be better compression schemes, but I can’t think up a better one so fast.
The largest hard-drive at the moment is 36 TB IIRC, so it can roughly store 2E+12 solutions. There are 18E+18 solutions so you need an array of 9 million of those disks. (and a nuclear plant as a power supply 
So the answer to you question is no, but …. if you could devise an algorithm that can generate the nth magic square on the fly, you do not need a big disk. There exists algorithms to generate the nth permutation but that is unfortunately not enough( by far). There are 36! permutations ~ 38E+40 so only one permutation in 2E+22 is a solution 