In a kingdom far, far away...

Let's start with this question - we may be unsure about how to say "away", but we do know how to say "far", right? ;) Right, it's רחוק!
⮞It goes too far! - זה הולך רחוק מדי! (audio #1)
⮞One doesn't need to go far to see the examples - לא צריך ללכת רחוק כדי לראות דוגמאות (audio #2)

But how do we say "far away"? Believe me or not, there is no word for for "away"... Just variations of the same ר.ח.ק. root...

In the sentence we started from, to give it a fairy-tale quality, we could change the word רחוק a bit and say:
⮞ בממלכה אחת, הרחק הרחק מכאן... (audio #3)

So, as for "staying away" - we just take the ר.ח.ק. root and make it into a reflexive verb - as a result getting להתרחק! :)

More examples:
⮞To stay away from the fire - להתחק מהאש. (audio #4)
⮞I looked at the boat moving away from me towards the horizon - הסתכלתי על הסירה מתרחקת ממני לעבר קו האופק.(audio #5)
⮞Stay away, you have no right to be here! תתרחק מכאן, אין לך זכות להיות פה! (audio #6)