The most common and most likely answer to this crossword clue is the 6 letter word HEIGHT
I've seen this crossword clue in The Guardian
H
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More clues leading to the result HEIGHT
Buddy explains!

The straight part
We know that "measure up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case measure up = height.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does HEIGHT mean?
HEIGHT noun- The distance from the base of something to the top.
- The vertical distance from the ground to the highest part of a standing person or animal (withers in the case of a horse).
- The highest point or maximum degree.
- A high point.
- An area of land at the top of a cliff.