"Measure" here refers to judgement. It is a noun. The specific thing that is being assessed is the spiritual value or maturity of other people -- remember when the apostles kept debating about who was greatest? That is the measurement being discussed here.
This continues Paul's theme of judgment, in which he condemns those who judge by the flesh, that is by what can be seen, in contrast with those who judge by the spirit, that is what cannot be seen with the eyes but is revealed by the spirit.
But the spirit will only reveal some things to you about some people, whereas the eye can see everyone. Therefore you should stick to your area - to what has been revealed to you, and refrain from trying to measure those that the Spirit did not give you insight about.
As what matters is in the heart, to properly judge someone you need to perceive their heart. The spirit can and does reveal the heart, if that person was put in your area of authority by the Holy Spirit.
But if that person was not put in your area of authority, what you will do is judge that person according to your own standards.
And what will be your own standards? It will be what is in your heart. The things your conscience condemns you with are what you will project onto others. So you will measure wrongly. You will be the body with the heart but everyone else will just be a body, onto which you project your own heart.
2 Corinthians 10:7 (KJV 1900)
Do ye look on things after the outward appearance? If any man trust
to himself that he is Christ’s, let him of himself think this again,
that, as he is Christ’s, even so are we Christ’s.
In other words, you are not the only one that Christ indwells. Christ is indwelling the person you are judging as well, and they have their own heart. When you judge wrongly, you are blaspheming God in them.
Paul was specifically complaining about those who judge him:
2 Corinthians 10:10 (LEB)
because it is said, “His letters are severe and powerful, but his
bodily presence is weak and his speech is of no account.”
The reason this was happening is that Paul reveals his heart in his letters, and so they see power. But when he visits in person, they cannot see the heart, because they judge according the outward appearance. So they see weakness. They then accuse Paul of being a phony or not living up to the outward appearance by which they measure him.
Paul's retort is that it is not he that is phony, but rather their failure to judge him correctly that is the problem. They are measuring wrongly.
Paul then condemns those who judge according to the flesh, and thus measure by only what is in their own heart, and not what has been revealed by the spirit:
2 Corinthians 10:12 (LEB)
For we do not dare to classify or to compare ourselves with some
who commend themselves, but they themselves, when they measure
themselves by themselves and compare themselves with themselves, do
This contrasts with Paul, who is able to measure them, because they were put in his area of authority by God as he is their spiritual father. Just as parents can "see through" their children, because they look at them with the eyes of love and concern, viewing them as their own flesh, so Paul has been given spiritual insight by God into the maturity of the churches he plants, and when he measures, he measures both rightly and according to the measuring rod given to him by God.
2 Corinthians 10:13–14 (KJV 1900)
But we will not boast of things without our measure, but according
to the measure of the rule which God hath distributed to us, a measure
to reach even unto you.
For we stretch not ourselves beyond our
measure, as though we reached not unto you: for we are come as far as
to you also in preaching the gospel of Christ:
Here Paul extends the metaphor of "measure" -- a measuring stick -- so that those things outside his spiritual insight would be outside the stick -- the stick wont reach them. But Paul is saying that his stick reaches all the way to them. If they were the spiritual children of someone else, they would be "other men's labors" and so beyond his measuring stick.
2 Corinthians 10:15–16 (KJV 1900)
Not boasting of things without our measure, that is, of other men’s
labours; but having hope, when your faith is increased, that we shall
be enlarged by you according to our rule abundantly, To preach the
gospel in the regions beyond you, and not to boast in another man’s
line of things made ready to our hand.
And Paul is saying that as the faith of the Corinthians grows, his stick will grow as well. They will never "outgrow" the reach of his measuring stick, but it will grow with them, and hopefully spread even to other regions.