(This is not to claim there is no such thing as the afterlife or the abolishment of death, Jesus addresses this in other words within Luke 20:35-36, which actually bears on this discussion later.)
The most direct indication that αἰώνιον ζωὴν does not mean "eternal life" in the way we normally think about it comes in John 17:3, where John uses a very specific grammatical construct to DEFINE αἰώνιον ζωὴν as "that they may know you, the one true God, and Christ whom you have sent."
Today I found another interesting testimony to this idea that when Jesus speaks to his Jewish audience of "αἰώνιον ζωὴν," He is not emphasizing or even referencing immortality but is rather referring to the question of who will be chosen to join in the blessings of the Messianic age. Check out John 5:39, shown in context below.
John 5:39–41 (NAB)
39 You search the scriptures, because you think you have eternal life through them; even they testify on my behalf.
40 But you do not want to come to me to have life.
This somewhat explains the question posed for Jesus in Luke 20:35-36 (see also Matthew's version in Matthew 22:23-31). The Jews assumed life in the coming age was similar to life in this one, which meant death, which meant the need to procreate, and hence meant that marriage was a necessary feature. But Jesus says that there is no death, and so there is no need for marriage, and hence their question is ill-posed.
So, the Jews do not believe in eternal life, yet Jesus accuses them of searching Torah because they they will find αἰώνιον ζωὴν there. This makes no sense if αἰώνιον ζωὴν is a reference to "eternal life." However, it makes complete sense if it is a general way of referring to the "life in the world to come." The Jews most definitely did believe in the World to Come, and they also believed that studying and applying Torah was the prime way to get there!
One statement from the Mishnah shows this in a backhanded manner: (Abot 3:11):
R. Eleazar the Modite says, “(1) He who treats Holy Things as secular, and (2) he who defiles the appointed times, (3) he who humiliates his fellow in public, (4) he who removes the signs of the covenant of Abraham, our father, (may he rest in peace), and (5) he who exposes aspects of the Torah not in accord with the law, “even though he has in hand learning in Torah and good deeds, will have no share in the world to come.”