Here's a sentence using "near" in the archaic sense of stingy:
Uncle Harold wouldn't even spring for a round of drinks, so cheap he bordered on near.
Note: This usage of "near" is not very common today. You might consider a more common synonym like "stingy," "cheap," or "miserly" depending on the desired emphasis.
An aviation term which has gained mainstream usage to describe the unfortunate event of two planes flying so close within an airspace that a passenger of one plane can be heard asking why the passenger of the other is getting steak for dinner.
Due to an aircontroller's ineptitude, the two flights to LAX were in a near miss situation.