In the early seventies, a man named Gary Kildall was experimenting with very small computers that didn't use paper tape. Instead, they used the early inexpensive magnetic media, which were much faster and had a higher capacity. Kildall wrote a new OS for machines that could boot off the new type of cheap magnetic media. His OS, CP/M, eventually became the premier OS for 8-bit microcomputers.
These were the disks (short for diskette) referred to in the names of the cheap CP/M knock-offs known as MS-DOS and PC-DOS. Earlier micro- and minicomputer OS'en booted from ROM or slow, expensive paper or magnetic tapes .
Contrasting the IBM-PC and other microcomputers of the 80s against even earlier micro's, one would expect "Disk Operating System" to refer to hard drives, but then it would have been a Disc Operating System. ...but that's another argument.