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For more on marking an answer as the "Best Answer", please visit our FAQ.spiff is correct. It is possible for space to be what physicists call "finite and unbounded" which means that it doesn't have any edges, but it doesn't "go on forever" either.
Imagine the Earth. Common sense tells us that (apart from local irregularities like buildings and mountains) the Earth is "flat". In other words, it looks flat; we go in straight lines, we travel sideways; we do not perceive any curves. But due to long-distance travel and scientific advances, we now understand that the Earth which seems flat is in fact a curved surface on the outside of a sphere. But the Earth is so big that we cannot get a sense of the curvature in our everyday lives.
If you travel in one direction on the surface of the Earth, what happens? You do not go on forever, but you eventually end up where you started from. If you squiggle around in different directions, you will eventually keep coming back top places you've been to before. Hence the surface of the Earth is a finite size (even though there aren't any walls to enclose it, or edges to fall off).
In physical terms, the surface of the Earth is a 2-dimensional surface of a 3-dimensional sphere. Similarly, we can say that the universe (the space we live in) is a 3-dimensional surface of a 4-dimensional hypersphere. So if you go off into space in a straight line, you will not go on forever, but you will eventually come back to the Earth from the opposite direction (without ever having turned round). It's a bit like travelling round the surface of the balloon that spiff was talking about. The dots on the balloon are like the galaxies in space - they are moving further apart as the balloon/ the universe expands.
And yes, the space itself (the vacuum) came into existence along with time and matter at the time of the Big Bang. It wasn't there beforehand (waiting for the Big Bang to happen inside it) because there was no "before".