It looks as if it is not clear what exactly the Great Disaster was.
According to Wikipedia (my highlights),
The 1975–1977 Hercules Unbound series and the OMAC backup stories in Kamandi and The Warlord tie OMAC to both the storyline of Hercules Unbound and to the Atomic Knights, indicating that the Great Disaster was the atomic war of 1986 that precipitated the events of the latter. Superman #295 (Jan. 1976) implied that the Great Disaster was a natural occurrence. DC Comics Presents #57 (May 1983) reveals that the events of the Atomic Knights stories were a fantasy in the mind of Gardner Grayle.
According to Comic Vine, (which isn't necessarily a canon explanation),
Very little is known as to the true causes of the Disaster, but what scant evidence exists points towards a mysterious energy wall known only as the Vortex. Earthquakes and tidal waves erupted all across the globe and the nations of Earth found themselves at the mercy of nuclear Armageddon. In the Countdown to Final Crisis, Karate Kid is stricken with the mysterious Morticoccus virus that is killing him and his fellow Legionnaire, Una, travels with him to find a cure. They end up on an alternate Earth where he dies and releases the virus all over the world. People begin changing into animals and the world is destroyed, essentially creating the world of Kamandi on the alternate world.
Then again, according to this article (which offers no source),
The Great Disaster, by the way, was started when Darkseid fought Zeus for the Anti-Life Equation. The presence of Ares causes World War III.
While nothing thus can be said for certain about the exact nature of the Great Disaster (and DC seems to have changed their mind on it several times), it does seem to have included a nuclear war and a mutagenic virus.