First look at the lyrics: http://songmeanings.com/songs/view/91758/
The references to "The eastern world is explodin', Violence flarin', bullets loadin'," are to the Vietnam war.
The words, "You're old enough to kill but not for votin'" refers to the fact that , in 1965, American men were subject to the military draft at age 18 and getting sent off to war, while at that time the minimum voting age (in all but four states) was 21.
"And even the Jordan River has bodies floatin’" refers to The War over Water.between Israel and its Arab neighbors over control of available water sources in the Jordan River drainage basin.
References to the button getting pushed and no running away, and not being able to save the world when it's in a grave, and the Eve of Destruction itself, are to the prospect of nuclear war between the US and Russia, where there were enough nuclear weapons to guarantee "Mutually Assured Destruction" - it would wipe out both sides completely and catastrophically, a terrifying prospect. At the time the song was written, there was a group of atomic scientists maintaining what was called the Doomsday Clock, that told how close to "midnight" = global thermonuclear war, the world was, based on international events and provocations. We were very close. The Doomsday Clock is still kept, but has changed in its scope.
The song also makes references to marches for integration (civil rights marches from Selma to Montgomery, Alabama), and Selma, Alabama where Bloody Sunday took place and peaceful marchers were attacked by police, and marchers were murdered.
"You may leave here for four days in space, but when you return it's the same old place" refers to the June 1965 space mission of Gemini 4 orbiting the earth, which lasted just over four days.
According to song's author, the lyric "The pounding of the drums, the pride and disgrace" refers to the Kennedy assassination.
The "hate in Red China" refers to the fact that we had no diplomatic relations with Communist China at all at the time the song was written. No trade, no talk, just hostility on both sides.
In short, it was a powerful protest song about the issues of the day that were putting the entire world in jeopardy - and pointing out that young people, who were being drafted and sent off to war, had no say in the matter ("old enough to kill, but not old enough for votin').