Who wants to live forever?

We see in the New Testament a few areas where Jesus is shown to be offering eternal life -- those who believe in Jesus have it. Or Paul saying that the wages of sin are death, but the free gift of eternal life comes from Jesus (or is something God offers to us through Jesus).

Yet we also have the concept of heaven and hell. Both hold the concept of an eternal existence. You either are eternally in Paradise, or eternally in a place that is rather unpleasant. Technically speaking, doesn't one have this eternal life regardless of where one ends up?

After all, that is what life is: existence. So wouldn't we logically say that when if God is offering someone eternal life, He is offering them the opportunity to exist forever? And if someone refuses this chance, that person will not in fact live forever? They'll cease to exist?

Except based on the heaven/hell theology, the person ends up in hell, which means the person is still existing. Does this mean we end up with a situation where while it does say that Jesus offers eternal life, "eternal life" means something other than the standard definition? Because if someone came up to me and said that s/he is offering me the chance to live forever, the implication is that I don't already possess the capability to live forever.