A POPULAR answer would be, Yes! The Scriptural answer is undoubtedly, No.
1. Because eternal life is a matter of promise.
"This is the promise that he hath promised us, even eternal life" (1 John 2:25). "Eternal life, which God, who cannot lie, promised before the world began" (Titus 1:2). "According to the promise of life which is in Christ Jesus" (2 Tim. 1:1).
The fact of eternal life being a subject of promise is proof that it is not the subject of present possession, for what a man has in his possession is no longer "promised" to him.
2. Because it is in the world or age to come that eternal life is to be received and enjoyed.
"He shall receive ... in the world to come, eternal life" (Mark 10:30). "He that hateth his life in this world shall keep it unto life eternal" (John 12:25). "God will render ... to them who ... seek for glory, honour, and immortality, eternal life ... in the day when God shall judge the secrets of men by Jesus Christ" (Rom. 2:7, 16). "The righteous shall go into life eternal" (Matt. 25:46).
If it is in the age to come that eternal life is to be conferred, it is manifest that it cannot be possessed in the present age.
3. Because "eternal life" means strictly "the life of the Age to come"; the word in the New Testament being derived from the Greek word for "Age". It is truly life which will never end, and therefore is also translated "everlasting life".
"They who shall be accounted worthy to obtain that world ... neither can they die any more" (Luke 20:36). "I give unto my sheep eternal life, and they shall never perish" (John 10:28). "The wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord" (Rom. 6:23).
If eternal life is everlasting life, it follows that in the present state we lack it, seeing our life is not everlasting, but comes to an end, necessitating our return to mother earth.
4. Because eternal or everlasting life results from a change of this corruptible and mortal body into an incorruptible and immortal one.
"He shall change our vile body, that it may be fashioned like unto his glorious body" (Phil. 3:21). "This corruptible must put on incorruption, and this mortal must put on immortality" (1 Cor. 15:53). "Clothed upon, that mortality may be swallowed up of life" (2 Cor. 5:4).
Our present mortal and corruptible state is therefore evidence that we do not now possess immortal life.
BUT YOU WILL SAY
"There are passages that plainly say we have eternal life," e.g., "Ye have eternal life" (1 John 5:13); "God hath given to us eternal life" (verse 11) "He that believeth on the Son hath everlasting life" (John 3:36).
True, there are such passages: but what are we to do with them? Are we to put such a meaning upon them as will contradict and make the other passages meaningless and useless? or are we to understand them in harmony with the other passages? There is only one wise answer: We must find the point of view that harmonizes them. The Scripture is at one with itself, and if we make it clash, there is something wrong in our interpretations.
You cannot make the idea of an actual present possession of eternal life agree with the passages which tell us it is a future thing: but you can make the idea of an actual future possession agree with statements that speak of it as a present possession. You ask, "How?" The answer is: By noting the custom of Scripture to speak of future things that are certain as though they were present. Examples:
"A father of many nations have I made thee" [when as yet Abram had no son] -- (Gen. 17:5).
"Unto thy seed have I given this land" [when as yet there was no seed] -- (Gen. 15:18).
"I [Isaac] have made him [Jacob] thy [Esau's] Lord and all his brethren have I given to him for servants and with corn and wine have I sustained him" [when as yet Jacob had realized none of these things] -- (Gen. 27:37).
There are many other examples. A single New Testament illustration may suffice: Jesus in prayer said to the Father, "The glory which thou hast given me, I have given them" (John 17:22), when as yet, even Jesus himself was not glorified (John 7:39). Jesus said, "I give my sheep eternal life". In the sense in which he gives it to them -- in promise and guarantee -- in that sense they "have" it -- not actually, but in the certainty of its future possession.
Christ now has it actually, and those who possess him possess it, for he is our life (Col. 3:4), and that life is eternal life, and this life is in Christ only (1 John 5:11). Christ dwells in the believer's heart by faith (Eph. 3:17). In this sense, and in this sense only, eternal life abides in him. But this sense is a very important sense, for the possession of eternal life by faith will lead to its actual glorious possession in the age to come.