Let's say I have just returned from a week's holiday. Can I say "I have been away for a week" or do I have to say "I was away for a week"? Or are both possible?
As is very often the case with the choice of perfect or non-perfect, both are possible, and the choice does not convey any difference in the objective events described, but just in how the speaker is choosing to relate to them.
I have been away for a week.
is more common, I would think, and emphasises the present relevance: either it is very recent, or I am still experiencing the effect of having been away, or it has consequences which are current.
I was away for a week.
is choosing not to emphasise that present relevance, but to treat the trip as something over and done with. This does not mean that none of the possibilities mentioned in the previous paragraph hold - they might do; but the speaker is not choosing to emphasise them.