Jesus speaking to the Jews: 39 You search the Scriptures because you think that in them you have eternal life; IMHO eternal life is a NT concept, it's (IMO) not in the OT. But apparently at the time of Jesus the Jews were looking into the OT for eternal life. There was awareness. Qeustions #1. Which possible OT passages the Jews could have meant? #2. What was the canon of that time? IOW, did the Jews get term eternal life from the Apocrypha of that time? Perhaps this is an unanswerable question? #3. Another hard question, how did the Jews of that time understood "eternal life"? Did they also believe back then that the soul is immortal? Applying 21th century logic, if the soul is immortal then everybody has eternal life, therefore if Jesus spoke about eternal life isn't not likely He was talking about the quality of (that) eternal life?
Here's the complete thought: v39 - 40 You search the Scriptures because you think that in them you have eternal life; and it is they that bear witness about me, yet you refuse to come to me that you may have life. You could take this a couple ways, but my understanding is that Jesus is saying a couple things: 1. The Jews He was speaking to rightly believed in eternal life 2. However, they wrongly believed that Scripture itself would grant them eternal life 3. In fact, the Scriptures 'bear witness' about Jesus, and He is who will grant those who 'come to [Him]' eternal life, but those He was speaking to refused to come to Him #1. Psalm 49, Job 19, Daniel 12, Isaiah 26, etc. (1 Maccabees?) #2. Depends on the Jew - remember that Paul got the Sadducees and Pharisees fighting about the resurrection, so there was no monolithic Jewish belief. #3. Pretty hard to use the word 'eternal' and mean something other than 'immortal', even then.
Job 19: 25-27 25 I know that my redeemer[c] lives, and that in the end he will stand on the earth.[d] 26 And after my skin has been destroyed, yet[e] in[f] my flesh I will see God; 27 I myself will see him with my own eyes—I, and not another. How my heart yearns within me!
Found an insightful article from the Jewish perspective how resurrection, eternal life, hell evolved. http://www.jewishencyclopedia.com/articles/12697-resurrection
After reading the mentioned article from the Jewish perception (and the various uncertainties) I get the feeling Jesus was exactly teaching that, if you believe in Me, you are immortal. Remains the logical question, those who not believe in Jesus are not immortal?