Here, Christ does not use the term "immortal" or "endless," to define eternal life, but He describes a kind and quality of life in terms of knowledge and a relationship with the Father and Son, a very intimate relationship.
Do not be misled by the limited Strong's definition of the word translated "eternal" in this verse. A more complete lexicon like Spiros Zodhiates' Complete Word Study Dictionary of the New Testament gives a more precise usage—how the word is used in the Bible rather than merely in classical Greek. Zodhiates says that the word refers to the "life which is God's" (p. 107). The life of God is more than endless, and that is what is important here. He adds, "It is to be understood as referring not only to duration, but more so to quality. That is, it is not merely life that is eternal in duration, but is primarily something different from the natural life of man, i.e., the life of God."
The Daily Bible Study Commentary: John (Volume 2) by William Barclay contains this comment:
There is another important thought in this passage, for it contains the great New Testament definition of eternal life. It is eternal life to know God and to know Jesus Christ whom he has sent. Let us remind ourselves of what eternal means. In Greek, it is aionis. This word has to do, not so much with duration of life, for life which went on forever would not necessarily be a boon. Its main meaning is quality of life. There is only one person to whom the aionis can properly be applied, and that is God. Eternal life is, therefore, nothing other than the life of God. To possess it, to enter into it, is to experience here and now something of the splendor and the majesty, and the joy, and the peace, and the holiness, which are characteristic of the life of God. (p. 207)
John 17:3 also contains the word "know." To understand eternal life, we must also understand how this word is used here. It undoubtedly contains elements of intellectual knowledge, understanding, discernment, information, and familiarity. However, this word suggests more than this because the Old Testament regularly uses "know" to describe sexual knowledge. Sexual knowledge between a husband and wife is the most intimate of knowledge. Husband and wife are no longer two but one flesh. In this regard, in John 17:3, the important thing is not the sexual act but the intimacy of heart and mind that in true love precede the act. To know God, therefore, is not merely to have intellectual knowledge of Him, but it is having an intimate, personal relationship with Him like the nearest and dearest relationship between two people.
Hosea 4:6 provides an interesting example of the practical effect of "knowing": "My people are destroyed for lack of knowledge. Because you have rejected knowledge, I also will reject you from being priest for Me; because you have forgotten the law of your God, I also will forget your children." God's implication is clear. If they had possessed knowledge of God, they would have had the power to avoid being destroyed. Nobody in his right mind wants to be destroyed. Ignorance might be bliss, but this verse shows it can be dangerously life-threatening.
Consider the implications of a lack of knowledge in the area of physical law. A person who does not know the power of electricity, nitroglycerin, carbon monoxide, drugs, or certain medications could pay for his ignorance with his life. Or, even if a person's ignorance of these things does not kill him, he might have the quality of his life severely impaired through a maiming, debilitating injury. However, when they are used with knowledge, they can do worthwhile things. Similarly, knowing God opens to men the freest and most rewarding expressions of an abundant life.
What if a person does not know of God's righteousness? Proverbs 11:6 says, "The righteousness of the upright will deliver them, but the unfaithful will be caught by their own lust." What a person does not have cannot deliver them, thus they are injured or destroyed. Isaiah 11:9 shows that during the Millennium, the knowledge of God will cover the earth like a vast ocean. This is what will make the Millennium so wonderful!
Eternal life is more than just endless life. The biblical eternal life includes power to produce quality living superabundantly far beyond merely existing forever.
We should touch briefly on its sexual aspect. Genesis 4:1, 17, 25 each contain the Hebrew word yada'. It has a wide variety of possible applications, one of which is "to lie by man." In each case in Genesis 4, it is translated as "knew," since that is its basic meaning. The Hebrews used it to describe the sexual part of the relationship between husband and wife; thus, it suggests intimacy. When applied to God, it highlights not merely being acquainted with Him but, as we would say today, being "inside His head." The corresponding Greek word, ginosko, translated "know" in John 17:3, can be and is used in the same way as yada' in Hebrew (see Luke 1:34).
To know God thus includes a wide range of mental, emotional, and experiential knowledge. The fruit of this intimacy includes love, reverence, obedience, honor, gratitude, and deep affection. We come to know Him as sovereign Ruler, Master, parent, brother, friend, Savior, and Lawgiver. We would never know this mixture of admirable qualities and authority without getting close to Him. They compel us to yield to Him with all of our heart while we strive to obey and glorify Him.
In summary, John 17:3 points to Jesus' indicating that eternal life is not merely endless, though that is its dominant sense, but that those who have it live intimately with God and conduct their lives as God does—otherwise, there would be no close intimacy with Him.