I tend to be of the camp which accepts the idea that eternity is one giant 4D object that extends from the birth of the universe to its death, all of which would exist simultaneously in 4D space, just as the top of an object in 3D space exists simultaneously with the bottom.
As 3D beings ourselves, our brains are only able to perceive one slice of the great 4D object at a time as we slide along it in our bubble of immediacy that we call "the Present". Otherwise, to experience the 4th Dimension in the way it truly exists (if indeed it is a solid object as mentioned) means you would experience your entire life from birth to death all in the same instant.
I had a friend once who said that our experience of time is like the universe giving itself an MRI, which sounds accurate, but seems to be a bit more crude and less poetic way to put it. I like to think that it's like playing a record on a turntable.
The entire album exists all at once on the record, but the music doesn't play until you add a "present moment" which moves forward along the groove - the needle. Poetically, in that metaphor I like to think of all of eternity existing at once to be like the record, our physical bodies to be like the needle, and our lives as we experience them to be the music.