The tree of life appears in two aspects in the Hebrew Bible. For me until now, as a Biblical Hebrew teacher, I was thinking only about the first one that is found in the Garden of Eden's story. The eternity is the option to live forever, the option of immortality, the outcome of eating two fruits, the forbidden one and the one that we couldn't reach.

When you read the story in Genesis 2-3, the tree of life is not the hero. We can find the tree of life three times in the story. The first time is in Genesis 2:9:

"וַיַּצְמַח יְהוָה אֱלֹהִים, מִן-הָאֲדָמָה, כָּל-עֵץ נֶחְמָד לְמַרְאֶה, וְטוֹב לְמַאֲכָל--וְעֵץ הַחַיִּים, בְּתוֹךְ הַגָּן, וְעֵץ, הַדַּעַת טוֹב וָרָע."

"And out of the ground made the LORD God to grow every tree that is pleasant to the sight, and good for food; the tree of life also in the midst of the garden, and the tree of the knowledge of good and evil."

The tree of life is found between two trees, The one Adam could eat and the one that he couldn't. The only information about this tree we can assume is its location in the garden. He is not an outsider; he is inside the garden without any doubt. Rashi mentioned in his commentary that the tree is the middle of the garden. Ramban added that the tree of knowledge was also located there and that the tree of life has fruits that gave long life and not eternity. One of the sages of Israel said in Genesis Rabbah that the tree could live for 500 years (the long life is actually for the tree and not for the man!)

When the story ends the tree appears one more time as written in Genesis 3:22-24-

"וַיֹּאמֶר יְהוָה אֱלֹהִים, הֵן הָאָדָם הָיָה כְּאַחַד מִמֶּנּוּ, לָדַעַת, טוֹב וָרָע; וְעַתָּה פֶּן-יִשְׁלַח יָדוֹ, וְלָקַח גַּם מֵעֵץ הַחַיִּים, וְאָכַל, וָחַי לְעֹלָם. וַיְשַׁלְּחֵהוּ יְהוָה אֱלֹהִים, מִגַּן-עֵדֶן--לַעֲבֹד, אֶת-הָאֲדָמָה, אֲשֶׁר לֻקַּח, מִשָּׁם. וַיְגָרֶשׁ, אֶת-הָאָדָם; וַיַּשְׁכֵּן מִקֶּדֶם לְגַן-עֵדֶן אֶת-הַכְּרֻבִים, וְאֵת לַהַט הַחֶרֶב הַמִּתְהַפֶּכֶת, לִשְׁמֹר, אֶת-דֶּרֶךְ עֵץ הַחַיִּים"

"And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever: Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken. So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life."

Here it state clearly that if the man would eat from the tree, he will live forever and would be as G-d and as his angels. Therefore, G-d prevents man of two things: the attendance in the garden itself and the arrival to the tree of life.

However, always the question of how can we live forever is asked.
At the same time, when Plato wrote his beautiful words and fables, sat down another wise man, according to the tradition that was Solomon, and wrote the same idea in the book of Proverbs.
In Proverbs 3:15 it is written:

"עֵץ-חַיִּים הִיא, לַמַּחֲזִיקִים בָּהּ; וְתֹמְכֶיהָ מְאֻשָּׁר"
 

"She is a tree of life to them that lay hold upon her, and happy is every one that holdest her fast"
 

'She' is the wisdom and the one who holds it is 'a tree of life'. From that verse, I understood that the wisdom has some kind of eternity. When you will learn Biblical Hebrew, you will see that is the difference between the definite article and the indefinite article. However, in this verse, it doesn't matter- the eternity remains!
The same idea, by the way, appears also in Proverbs 11:30-

"פְּרִי-צַדִּיק, עֵץ חַיִּים..."
 

"The fruit of the righteous is a tree of life…"
 

When a person is wise and righteous he lives forever, even if he is not with us. His ideas, his actions and his behavior were, are and will be a model to us!