'Eternalism' is the English translation of the Pāli terms sassatavāda or sassatadiṭṭhi. The former indicates the profession of a doctrine of eternalism, indicated by the word vāda meaning 'speech' or 'words'; while the second is the eternalist view or belief, using the word diṭṭhi 'see', or 'view' and by metaphor an 'idea'. In practice they are more or less synonymous. Sassata, therefore, should mean something like 'eternal'.
The Pāli English Dictionary (PED) suggests that sassata comes from the Vedic śaśvat, and means 'eternal, perpetual'. So far I have not found a convincing etymology of the word śaśvat. It is used in some form in the older books of the Ṛgveda, but not often. Of the words that begin with śaś- the vast majority are to do with hares or rabbits, or with the moon. Ancient Indians saw a rabbit in the moon and so named it, amongst other things, Śaśin - literally 'with a rabbit'. The root śaś is said to mean 'leap' and a rabbit is a leaper. These other words however all suggest something recurring: śaśaya, śaśīyas, śaśvadhā, śaśvāya, śaśvata, śaśvatika. Now one of these in particular gives us some information. Śaśīyas is a comparative in which the suffix -īyas is added to an adjective. The dictionary tells us that śaśīyas means 'more numerous, mightier, richer'. This suggests that there is an adjective śaśi or śaśī meaning 'numerous, mighty, rich', and a verbal root śaś. However neither is in the dictionary, or in the Whitney's list of verbal roots. As one friend said - not every word can be traced to a verbal root, and śaśvat may be irreducible. If there were a root śaś then the suffix -vat might be a possessive, and śaśvat may be conceived of as something like possessing numerousness. Etymology not withstanding the basic idea conveyed by śaśvat seems to be recurrent rebirth and redeath (punarbhava and punarmṛtyu).
Skeat's etymological dictionary relates "eternal" to the Indo-European root √i 'to go' which occurs in Sanskrit as the verb 'eti'. The same verb occurs in Pāli, and in the imperative may be familiar from the phrase ehi bhikkhu - "come monk" - with which the Buddha indicated that he had accepted someone as a disciple. The word comes into English via the Latin word ævum meaning 'age' or 'life period'. Eternal, age and aeon are related words.
This illustrates that when we translate we need to sensitive to nuance, because eternal suggests to an English speaker something which is 'always on'. And unending process or state. Whereas the India word suggests something which happens again and again, an recurrent process and a repeating state. Thought these two concepts are related, they are not identical. However eternalism is so established as the translation of sassata that I doubt it will change because I say so!
Let us turn to what eternalism is according to the Pāli Canon. We do not have to go far because sassatavāda is defined near the beginning of the first Sutta in the Canon - the Brahmajala Sutta. The eternalist doctrine is said to be applied in two ways: to the ātman and to the loko. I think that this is one of those occasions where the word 'loka' means the material world. We might translate it here as 'universe'. There are four ways that one can come to hold the sassata doctrine. The first three of these are of most interest to us. Firstly a person may, through meditation, experience recall of very many previous lives. And they may conclude that this process simply goes on for ever. Or they may, again in meditation, see either a few, or many world ages (these are two of the four cases). A world age is the almost infinitely long period in which the universe is created, persists, and is destroyed in a conflagration. Having seen these repeat themselves in visionary meditative experiences a person may come to believe that they just keep happening without end. The fourth involves reasoning - one simply comes up with a metaphysical theory. These kinds of theories are not all bad - the theories of physics and chemistry for instance are useful even though we know them to be only models - but in this case the theory is not helpful.
The idea that the ātman comes back again and again is a view from the early phase of Upaniṣads. Already in the later parts of the Ṛgveda there are hints that the Brahmins believed that having died here one went to the fathers (pitṛ) there, but after a time returned to live here again. This process seems to have been endless. In the Bṛhadāranyka Upaniṣad (BU) the idea undergoes a refinement which Johannes Bronkhorst associates with the śramaṇa group known as Ājivakas. We mainly know about the Ājivakas from the polemics against them in Buddhist texts. In the first chapter of BU there is a form of rebirth which is a cycling between this world and 'the other' and it is the ātman which journeys between the two. Note that here there is no sign of a way out of the cycle - no liberation or mokṣa. Later on again the Brahmins adopted the idea of mokṣa. Note that the reference to this world and the next is commonplace in the Pāli texts - which suggests that this version of rebirth remained current alongside other theories, for sometime.
Why is a view that the ātman or the universe is eternal a bad thing? It is because there is no possibility of escape from the cycle. Escape from the cycle, liberation, mokṣa was a crucial aspect of the what the Buddha offered to his followers. It is likely that the Buddha was not the first to talk about escaping the round of rebirth, that the Jains and Ājivakas also believed that escape was possible which is what motivated them to do severe austerities. But clearly some people in the Buddha's day still believed in this relatively unsophisticated view of rebirth and did not accept that liberation was possible. If we do not believe in liberation we won't be motivated to practice with the intensity required for liberation. So the eternalist belief is counter-productive.
You may be wondering why these ancient metaphysics are still relevant. If we believe in rebirth at all, we certainly don't literally believe in an endless cycle of rebirths. Western Buddhists tend to take on views of rebirth consistent with whatever Buddhist tradition they follow, and they are forewarned against eternalism. It turns out the any kind of eternalistic belief denies the possibility of liberation, any kind of unchanging entity fouls up the logic of it. And we do tend to have eternalistic beliefs. The whole world view of Christianity is based around ideas of an eternal God, and an eternal soul; eternal heaven, eternal damnation. Salvation is not based on being liberated from suffering, but from an acceptance of Christ leading to eternity in heaven. Many Western Buddhists, even those not overtly bought up as Christians, will have a view like this lurking somewhere in their psyche. More over the idea of an eternal soul plays along with the human tendency to see ourselves as continuous over time, and going on with out end - we think we're going to live forever and so we don't face the world as though our time here is limited and precious.