The requirements for magnetic core properties are really well
known; old JCC's, PROCEEDINGS and lots and lots of textbooks
discuss it reasonably deeply.
One word: hysteresis.
Successful commercial core memories require grossly underpaid
philipino housewives or other exploitable labor; welcome to the
fruits of capitalism.
I bought a few hundred thousand new/unused plain cores on ePay a
few years ago, for about twenty bucks. The problem is they're
about .005" OD! If you're going to make your own at home without
slave labor to go blind for you, you'll probably want cores large
enough to handle.
If you just want to play and make a 4 x 4 core or something, I
wonder if you couldn't get decent hysteresis with some other
ferrite product. You could compensate for a "poor" core with good
electronics and/or brute force. I haven't looked at a toroid spec
sheet in ages and not for hysteresis.