It marked this wrong for me
What about "nearby"?
Yeah, it even gives that as a translation when scrolling over the characters, but marks it as wrong.
Yep, report it!
Can't this be translated as "I am not near"?
Are all of these more common ways of speaking than using ありません
Traditionally -くありません is the most correct way to negate い-adjectives, but I think that -くないです is used much more common nowadays.
ない translates to "not". ありません is the negative form of あります meaning it is more formal. ない is more common but good to know, especially for words like these that sound wrong using ありません.
Why does "it is not near here" work?
I think that would be ここからちかくないです。
it's already been mentioned but "nearby" should also be accepted.
近くないです（chikaku nai desu）