Question: The Big Bang theory, which is widely accepted, indicates that existence did emerge from non-existence. But why do Objectivists hold that existence is forever?
Answer: There are four points that need to be made in response to your question.
First, "non-existence" is not the name of something. "Existence" is our name for the totality of all that there is. "Non-existence" is a concept that does not refer to anything; it can be validly used only as an abstraction in stating negative propositions (as in "God does not exist"). Objectivism rejects the idea of "existence emerging from non- existence", because it treats "non-existence" as a kind of thing, something that can act and have consequences. This is the fallacy (also committed by Existentialist philosophers) which Ayn Rand called "the reification of the zero" (she discusses this fallacy in chapter 6 of Introduction to Objectivist Epistemology).
Second, time is the measurement of change; it is the relationship between different changes occurring in various existents (for example, when you measure the duration of some event in days, hours, minutes or seconds, you are relating it to the circular revolution of the earth). Time therefore depends on the existence of entities that change; it is time that is within existence, not the other way around. The idea of existence as a whole starting at some point in time, is self-contradictory, because it requires there to be time before existence, i.e. time without any entities that change. That is why Objectivism rejects this idea, and regards existence as eternal, i.e. outside of time.
Third, it is not true that Big Bang theory holds that existence started at some point in time, or that existence emerged out of non-existence. Rather, it holds that something very different from matter as it exists today, and perhaps operating by physical laws different from physical laws as they are today, existed before the Big Bang, and caused the Big Bang (this pre-Big-Bang existence is usually referred to as "the Primeval Atom"). While some have interpreted Big Bang theory as holding that existence emerged out of non-existence, or as confirmation of the religious idea of creation by God, these are fringe interpretations. Mainstream Big Bang theory, as widely accepted, does not in any way challenge Objectivist ideas.
Finally, we should understand the relation of philosophical axioms to scientific theory. Unlike many philosophical systems in history, Objectivism does not have a cosmology; it does not have positive theories about the nature of the universe. The Objectivist metaphysics is limited to identifying the fundamental, self-evident axioms - such as the concepts of existence and of time - which are the pre-conditions of any attempt to understand the universe - and specifically, the pre-conditions of all scientific investigation. (Rand discusses this also in ch. 6 of Introduction to Objectivist Epistemology.) Any attempt to challenge these philosophical axioms based on a scientific theory is self-contradictory, committing "the fallacy of the stolen concept"; it challenges the axioms based on complex ideas and concepts that depend on these axioms. This means that in any case in which a scientific theory appears to challenge the philosophical axioms of Objectivism , you can be certain that either the theory is invalid or it is misrepresented (and in the case of Big Bang theory, the latter is true).