Far and far away technically have pretty much exactly the same meaning; however, because far away has two words instead of one, it generally has a bit stronger feeling to it (this is true in general: The more words you add to a term or phrase, the stronger the impression), and thus is usually used to describe longer distances. Far away can also sometimes have the implication of "out of reach" in some way.
So if you're talking about the difference between going down the block vs. across the city, you would probably just use far. If you're talking about the difference between going to the next town over, or across the continent, then you might use far away instead:
I was thinking of going to a restaurant, but I can't go very far.
I need to choose a university to attend, but I don't want to move very far away.
For the situation you described, your answer ("I can't go very far") is perfectly fine. Saying "I can't go very far away" is also OK (it's not wrong), but it's just not what most people would probably say in that case.