At the time, many computers were decimal, or had word lengths like 18, 24, 36 or 48, or even 72 bits. Characters were usually 6 bits. The power-of-two standard based around an 8-bit byte didn't exist yet.

Whoever picked 60 bits (Cray?) was almost certainly thinking in octal, not hex. 60 bits is multiple of 3, it fits in 20 octal digits, and it holds 10 characters. Most importantly, a 60-bit floating point number is precise enough for just about any calculation.