Scientifically and logically speaking the concept is unknown or perhaps unknowable.
Yes, in our modern world of Euro-centric domination wherein our understanding and knowledge of science and philosophy have become confined to the Western narrative of history, — a systematic bias deeply entrenched in Western/Westernized academia as well — most theological concepts such as 'eternity' appears to have remained unsolved and unexplained.
However in the largely unknown world of muslim philosophy — especially with the major conclusive strides made by the late theosophist Mulla Sadra — most theological concepts have now been effectively resolved and logically explained including God and His attributes, heaven and hell, soul, angels, Satan, and Hereafter, among others. In this answer (which is the first of its kind on this site), we will hopefully testify to the richness and capability of muslim philosophy in defense of Islamic tenets.
God and His Attributes in the Light of Being and its Attributes in the Philosophy of Mulla Sadra
Eternity, based on Mulla Sadra's Ontology of Being, should be seen as one of the essential characteristic or attributes of Being. The simple concept of Being for Sadra, when considered by the virtue of its very essence, implies full perfection in every possible respect. That's because according to Sadra (and in fact all followers of the Neoplatonic tradition), all imperfections are attributed to lack of being or existential deficit.
Absolute perfection for Being — that is, for Being to be fully being or distilled of any lack of being — requires it to be simple — that is, non-composite, unified, non-discontinued and fully connected —, all-pervasive and unique. Being also constitutes every and all beings. For if it doesn't constitute some beings, it contradicts its very essence as "all beings and altogether being."
When Being is conceived as such, it follows that whatever we conventionally perceive and think of as beings or things (i.e. the plurality of phenomena within and without us) are only different manifestations of Being and therefore have no sovereign and independent reality from that which constitutes their very existence, Being.
As for the epistemological status of Being, while everything, that is, all manifestations of Being can be demonstrated by observation and be scientifically or philosophically defined, Being is so evident and so readily intelligible that it is needless and beyond any definition. Also, while the manifestations of Being are perceived separately, distinguished and defined due to their very existential limits and deficits, Being due its unified, single and all-encompassing reality can only be distinguished from non-being while lying at the core of every observable or logically demonstrable being.
For Mulla Sadra, Being conceived as such corresponds to the reality of God. God similar to Being exists in everything at the same time can not be equated or reduced to any thing. Since Being is the source of all perfection, hence the corresponding Islamic belief that God is the source of all goodness. In short, Being in philosophy is basically identical to God in religion, expressed only in different terminology.
Now to put this into the relevant Islamic theological perspective, Let's summarize the discussion as such: Everything that is, owes its share of being to Being, while the imperfections that are a result of the relative lack of being of things are not attributed to Being but evidently to non-being. This is the basis for a famous statement by Sadra and his preceding theosophists that: the inclusive simplicity (of Being) is 1) all things and at the same time 2) none of the things; respectively corresponding to the idea behind Quranic descriptions of Allah as in the three opening verses of Surat al-Hadid implying Allah to be virtually everything; and the descriptions pointing to the doctrine of Tanzih as in Surat al-Shura:11 that says "there's nothing like him".
Therefore when the concept of Being or God is properly understood as such, its essential attributes such as eternity become easy to relate to as well. Since Being or God is all being and altogether being , it can not hold any attribute that violates its very essence such as spacial or temporal mode of existence that characterizes beings (that is, creation) for these attributes are rooted in lack of being or existential deficit. In other words, anything confined within one location or one period of time, necessarily lacks the being of other things positioned in other modal circumstances, and therefore can not represent Being which is "all beings and altogether being."
With respect to what was said about ontological status of Being, the essential attributes thereof (such as eternity) can not be sensually observed or logically defined either, because of the utmost a priori self-evidence of Being and its attributes. Therefore the problem of correct comprehension of such attributes arises only when people want to wrongly understand them in terms of some of the beings (i.e. creation) which by the virtue of their deficient existential status can not posses those attributes of perfection. This is the distinction that should be always kept in mind when contemplating attributes that are unique to God.