FROM V.V. RAMAN, EMERITUS PROFESSOR OF PHYSICS AND HUMANITIES AT THE ROCHESTER INSTITUTE OF TECHNOLOGY: Eternal life, as a mere persistence of life here on Earth, could be very boring. But then, considering that old age doesn’t have to be boring, if there are things to do and experiences that one enjoys, whether it's listening to music, eating chocolates, reading romance novels, or tending to gardens, never-ending life could be a welcome permanence.
We may trivialize the concept by saying that it all depends on the phase and health of life at which this imagined eternal existence is going to be. If it is going to be in the sort of physical body we are accustomed to in the prime of salubrious life, with refreshing showers, decent wardrobes, and variation in breakfast cereal and dinner menu, with near and dear ones within reach, it would be ideal.
We may say that our images can be eternal (whether as fading photographs or on YouTube), by which we mean existence of form and sound for others to see and hear as long as technology and terrestrial civilization last. Likewise, there is immortality in the ideas and discoveries left behind. In recent decades, some physicists have argued that eternal life is a distinct possibility for us all in cyberspace. Like alchemists of ancient times, one might concoct any theory to make ourselves live in saecula saeculorum: forever and ever, to use a biblical phrase from Galatians.
The temporal view of eternity was expressed by the poet Bayard Taylor:

Till the sun grows old,
And the stars are old,
And the leaves of Judgment Book unfold.

Eternal life is a sophisticated theological concept, which has little to do with physical time. It is related to the doctrine that we are all endowed with a soul. The soul may be envisioned as an intangible supernatural entity that can exist beyond space-time, with an innate connection to an Eternal Being. The general religious belief is that in the postmortem phase, eternal life would be ecstatic for the soul that attains the celestial realm.
The eternity of which religions speak is holistic existence that transcends ticking time. As Robert Neville put it in his book Eternity and Time’s Flow, “eternity is the togetherness of past, present, and future in which they are all equally real and in which each allows the others to be what they are precisely in their temporal difference, … eternity is an eternal togetherness.” In the imagery of poet John Donne, who described time as a short parenthesis in eternity, we may say that our physical life is an even shorter parenthesis within that parenthesis. Thus, eternal life would involve jumping out of two parentheses into an ocean without bounds. Another poet noted that in mystic moments, we are watching the shadows of eternity. In this metaphor, we may say that eternal life would be moving from sheltered shade to dazzling sunshine.
From a mathematical perspective, the only things of whose eternal life we can be rationally certain (not in the temporal sense but in their continued actualization ad infinitum) are the values of pi and other transcendental numbers that go on and on and on without end in their decimal mode, and the integers more generally (1, 2, 3, …). But their eternal life, impressive as it may be, gets to be pretty drab after a while.

V.V. Raman appears with Richard Swinburne, J.P. Moreland, Seyyed Hossein Nasr, Neil Gillman, Michael Tooley, and Huston Smith in "Eternal Life Is Like What?" the 38th episode in the Closer to Truth: Cosmos, Consciousness, God TV series. The series airs on PBS World (often Thursdays, twice) and many other PBS and noncommercial stations. Every Friday, participants discuss a recent episode.