From a physical standpoint:

Physicists and cosmologists tell us that time and matter came into being simultaneously at the moment of the Big Bang. Therefore, time in our universe is not infinite in both directions. One can go in the direction of the future for an infinite length, but in the direction of the past one can only go to the point of the Big Bang. Nothing existed before then, time included.

Of course, that presupposes an open and infinitely expanding universe, not a closed universe which will eventually contract back upon itself. If the universe is a closed system which expands and contracts in cycles, then our Big Bang could be one of an infinite number of Big Bangs. In that case, it could be said that time is essentially a closed loop. It's created at the Big Bang, ceases to exist when the universe once again collapses into a singularity at the Big Crunch, and is created anew with the next Big Bang.

From a metaphysical standpoint:

I really hate it when eternity is spoken of in a religious or spiritual context. Eternity is a measure of an infinite amount of linear time. Linear time is a feature of our physical universe. To speak of any state of being outside our physical universe as "eternal" is to say that said state of being is defined or constrained by linear time. There's a logical inconsistency there. Any state of being that exists independently of the physical universe must also exist independently of linear time. It would not merely exist within time unto eternity, it would exist outside of time altogether. There's a big difference, and when I hear some preacher at a funeral going on about "eternal life" I always wonder whether he's thought about the matter as deeply as he pretends to have.