The phrase "everlasting life" and "eternal life" comes from two Greek words, the adjective, aionios, which doesn't quite mean "eternal," and the noun, zoe, a word meaning "living" both in the sense of existence and "making a living." These words appear together eighteen times in Jesus's words as ζωὴν αἰώνιον, a phrase translated as "everlasting life," "eternal life," and "life everlasting." This idea has a very specific meaning in modern Christianity as life in heaven.
In this article, we will discuss how people of Jesus's time would have heard them. Jesus uses it to mean the "existence" of physical life, spirit plus body. For more on how Christ uses this word with other words about human existence (soul, heart, spirit, body, etc.), read this article. However, they wouldn't have thought of this life as "eternal" as much as "on-going." Jesus defines this phrase as to "keep living" or "maintain life" in John 5:40 after using it in John 5:39.
What Kind of "Life" is Zoe?
The noun, zoe, means "living," "substance," "property," "existence," and, incidentally, "the scum on milk." It has the sense of how we say "make a living" to mean property. Homer used it more to mean the opposite of death, Jesus uses it forty-two times to refer to life in various ways. He seems to use it as physical life in a body, but only once in opposition with the word for "death," and that verse uses the phrase "everlasting life." For more on how Jesus uses this word with other words about human existence (soul, heart, spirit, body, etc.), read this article. In that article, we discuss a different Greek word, psyche, that is also translated as "life" but which has more the sense of internal life--conscious awareness and memory.
Zoe is the life that comes from our "substance,” by which the Greeks meant our personal property, as well as our physical existence. The thinking was that we spend our time living working to survive, generating property. In the verb form, it is a metaphor for "to be full of life," "to be strong," and "to be fresh.” The basic concept is that this "life" union of flesh with spirit, the "breath of life," pneuma. Jesus specifically says that this pneuma is what gives life in John 6:63.
The word zoe also implies a productive life, one that earns its keep. Our ability to life in the sense means our ability to earn a living, maintain our bodily health by finding food, drink, shelter, and so on.
How Everlasting is Eternal?
"Everlasting" is aionios, which means "lasting for an age," and "perpetual." It has come to mean "eternal" in Greek, but that concept arose from the Bible itself. It is an adjective from the Greek word, aion (αἰῶν) which means "life," "lifetime," "age," or "generation." This word is also translated in the Bible as "world," but it only means "world" in the sense of the current era of the world. See this article on words translated as "world" in Jesus's words.
Notice that the root word aion means "life" in the sense of "lifetime." This word does not mean "forever" or anything like it. It specifically means a limited span of time. It was a measurement of time like we talk about "a generation," that is, the space between one generation and the next, which is around thirty years. The word aion can refer to longer period of time, but it is not the word for "eternity" or anything like it. In modern Greek, this word has become the basis for the Greek words for "eternity," aioniotes (αιωνιότης) and aionioteta (αιωνιότητα) but not in ancient Greek. These concepts were taken from the Bible, making its way into modern Greek.
The ancient Greeks used the words "for all ages," using the plural of aion, to come closer to describing"eternity," or more simply "for ages." Another word, athanatos, (ἀθάνατος) literally meaning "not dying" is much more commonly used to express this idea as "undying" and "immortal." This word was very common in ancient Greek, used to describe the existence of the Greek gods. However, we must remember that even the Greek gods were not seen as lasting for eternity. They could be destroyed even though they did not die. The gods themselves had destroyed their own parents, the Titans.
The ancient people in general did not have the same clear idea of "eternity" that we seem to have now. They saw the world as constantly changing. Nothing was unchanging, not even the gods who could be destroyed and changed. From my study of ancient Chinese, I never ran into the concept. Indeed, much ancient Chinese philosophy holds that nothing can last for all time or outside of time.
We also see this inability to express the idea of "eternal" among ancient Judeans. The word "eternity" only appears once in the KJV translation of the Bible, in Isaiah 57:15. The Hebrew word is 'ad, עַדI which means "perpetually," and "into the future" is translated as "eternity." However, more generally, it has the sense of "ongoing," which is probably a pretty good way to describe the Greek aionios. In the Greek Septuagint, this was rendered as "an age of holy awe," which is kind of a cool way to describe a long time. Despite the focus on "eternity" in more modern Christianity, this is as close as the KJV translators could honestly come to finding the word in the Hebrew books.
The concept of "eternal life" is a religious dogma that has arisen from the teachings of Jesus, but it wasn't really a part of them. Jesus promises only a "life after life," that is, an on-going life. This is not a life outside of time or beyond time. It is a life that continues to its next stage, whatever that might be.