In John 17:3, as Jesus prays, he says,
And this is eternal life, that they know you, the only true God, and Jesus Christ whom you have sent.
What Christ is saying is that the eternal life that we receive by faith in him is more or less defined by the new relationship that we have with the Father and the Son. We know God and his Son Jesus Christ, and that in in itself somehow is eternal life. This life needs to be understood as both a quantity and a quality of life.
First of all, eternal life can be described everlasting or never-ending or using some other such term that means it will just keep on going and will not stop. I emphasize this point first because I sometimes hear Christians focus on eternal life as a quality of life as opposed to a quantity of life. I understand why. It is possible to misunderstand eternal life as being much like the life we have now, complete with its troubles and our imperfections, just without end. That is surely not what Jesus means. But if we say it is a quality of life rather than a quantity, then we are not doing justice to the term eternal, for which Jesus could have substituted some other term unrelated to time (most excellent or super wonderful?) Why confuse us by calling it “eternal” if he didn’t mean for time to be involved at all?
So this seems to be a false dichotomy. It is not an either/or situation; it is a both/and. God is eternal; that is one of his attributes, but we are not, since we are created beings. He can, however, give us life everlasting if he wishes, and he does if the Bible is to be believed. This life is acquired by an intimate connection with him, facilitated by the Holy Spirit who regenerates us and gives us new life in Christ.
Second, we must speak of this new life as qualitatively different from the mere earthly life we were born with. We have come to know the only true God and Jesus Christ, his Son, the One sent here for our salvation. Prior to this we did not know God at all. We may have known of him, in a fact-knowing kind of way. We may have believed that there was a God in a less personal way. This is often true of people who later come to know God personally. They come to believe in God in the abstract on the way to a personal faith. It was certainly true in my case. But to know God personally and to know him as our Father is a very different thing indeed.
If we really know God and his Son Jesus Christ, we can no longer live as if that does not matter. We need to bask in the warmth of their fellowship. We need to learn from their wisdom and rejoice in their love. Life can never be the same. Eternal life is ours and this life lived in the very presence and power of God will never end.