I assume it was in their head.

I don't know exactly how large of a times table I have in my head, but it's larger than 10x10, more like 15x15. A 60x60 table is 16 times larger than that, but I also learned my times table in elementary school and being a scribe or whatever the job title was for a guy who multiplied was very likely a full time job for a grown Babylonian. Learning your times tables up to 60x60 seems totally plausible.