That is a great protest song by P.F. Sloan, sung by Barry McGuire and written in 1964.
Here are two verses taken from the song.
"Eve Of Destruction"
The eastern world it is exploding
Violence flarin', bullets loadin'
You're old enough to kill but not for votin'
You don't believe in war but whats that gun you're totin'?
And even the Jordan River has bodies floatin'
But you tell me
Over and over and over again my friend
Ah, you don't believe
We're on the eve of destruction
Don't you understand what I'm tryin' to say
Can't you feel the fears I'm feelin' today?
If the button is pushed, there's no runnin' away
There'll be no one to save with the world in a grave
Take a look around you boy, it's bound to scare you boy
And you tell me
Over and over and over again my friend
Ah, you don't believe
We're on the eve of destruction
It was written in 1964, but this song has just as much validity today as it did back then.
Nuclear missiles and bombs were relatively new to warfare. War between Russia and the USA seemed a distinct possibility over the Cuban missile crisis. The Vietnam war was just getting started as far as the USA was concerned. Problems were escalating in the Middle East between Israel and Arab nations. American race riots were a few years away.
"You don't believe we're on the eve of destruction"
The writer is outlining huge problems of the time. He cannot understand why others (mostly the older generation then) cannot see that nuclear war and anarchy is just around the corner.
The "eve" is the end of the day. it is short for "evening". In the writer's view the day was full of conflict around the world and it was ending and when the new day comes the world could be destroyed (metaphorically).
Of course, we dodged that bullet back then after a decade of problems. However, if you read the lyrics they still have meaning today. This was a protest song that 50 years later still rings true. Many of those same problems still exist. The only difference between then and now is there does not seem to be as much war protest by youth of the world.