Karl Barth appeared on a radio program called "What Do You Think, Professor?" near the end of his life (1960's), where he answered bible and theology questions similar to Hank Hanegraaff's old famous radio program "The Bible Answer Man". A caller asked Barth if in light of parapsychology, interaction with the dead was possible, and this spurred a conversation about eternal life.
Afterwards, Barth received an angry letter from Werner Rüegg in Zurich, expressing that he was dissatisfied with Barth's answer about Eternal Life. Barth responded that he was strictly answering the parapsychology question, and intentionally did not adequately explain Eternal Life. Karl Barth responded to Rüegg with the following fuller and fascination explanation of Eternal Life, which includes a reference to a hymn by Gellert (that I've included as an appendix).
Eternal life is not another and second life, beyond the present one. It is this life, but the reverse side which God sees although it is as yet hidden from us—this life in its relation to what He has done for the whole world, and therefore for us too, in Jesus Christ. We thus wait and hope, even in view of our death, for our manifestation with Him, with Jesus Christ who was raised again from the dead, in the glory of not only the judgment but also the grace of God. The new thing will be that the cover of tears, death, suffering, crying, and pain that now lies over our present life will be lifted, that the decree of God fulfilled in Jesus Christ will stand before our eyes, and that it will be the subject not only of our deepest shame but also of our joyful thanks and praise. I like to put it in the fine stanza of Gellert in which he speaks of knowing in the light what is now obscure on earth, of calling wonderful and glorious what took place inscrutably, of seeing with out spirit the context of our destiny with praise and thanksgiving. 
Barth explains that Eternal Life is not a future life beyond death, such as an afterlife, but it is another view of this life we live in retrospect. Barth acknowledges that our lives are full of "tears, death, suffering, crying, and pain", but he believes that one day in the future, on the last day, the coming of Jesus will usher in the Final Judgement, and this judgement isn't a trial that determines whether we will be in heaven or hell for unending time, this judgment will allow us to understand our past lives differently like a paradigm shift. Jesus is already the savior of the entire world, but the lordship of Jesus is now hidden from us, and we are unable to see God's redemption through the horrors of this cursed world. However, Barth beliefs that that the final form of the coming of Jesus, will lift the veil and give us a new understanding and vision of our past lives, and we will be able to understand out lives in a new way that we cannot understand now. Barth explains that the coming of Jesus at the end of time, will allow us to see the glory of God in our lives, such that great and deep shame of our sin will revealed, but also our deliverance from our sins, by the death and resurrection of Jesus Christ, and this will causes us to experience the fullest joy and thanksgiving.
Fourth verse of the hymn: "Nach Einer Prüfung Kurzer Tage" (After A Trial Lasting A Short While) by Christian Fürchtegott Gellert (1715 - 1769) :
|Dann werd ich das im Licht erkennen,
Was ich auf Erden dunkel sah,
Das wunderbar und heilig nennen,
Was unerforschlich hier geschah,
Dann schaut mein Geist mit Lob und Dank
Die Schickung im Zusammenhang.
|Then [in eternity] I will will recognize in light
What on earth I saw but darkly,
I will call wonderful and holy
What happened unexpectedly here,
Then look upon my spirit with praise and thanks
The connections in these acts of God
[^Header Image] "Karl Barth." N.p., n.d. Web. <http://barth.ptsem.edu/uploadedfiles/bigsectionphotos/barth-1967.jpg>.
[^1] Barth, Karl. "#4 To Werner Rüegg, Hombrechtikon, Zurich Canton, Basel, 6 July 1961." Karl Barth Letters 1961-68. Trans. Geoffrey William. Bromiley. Grand Rapids, Mich: Eerdmans, 1981. 9-10. Print.
[^2] "Nach Einer Prüfung Kurzer Tage" (After A Trial Lasting A Short While) by Christian Fürchtegott Gellert (1715 - 1769). For more information, see: http://www.lieder.net/lieder/get_text.html?TextId=93501
Related: Eternal Life, Karl Barth, letters