"Wherefore of these men which have companied with us all the time that the Lord Jesus went in and out among us, beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection" (Acts 1:22-23).
From what I read, there are two requirements:
1) company with Christ during His earthly ministry until His ascension, and Even though Paul was not in the original 12, or probably not even present when they chose Matthias, Paul cited it in defense of his own apostleship: "Have I not seen Jesus Christ?" (I Corinthians 9:1).
2) witness of the resurrected Lord. These were the qualifications which had to be met by the new twelfth apostle, who, as it turned out, was Matthias. Clearly, this eliminates any present day apostleship.
My understanding from these verses is that no man can be an apostle who has not been a witness to the risen Lord.
I have met people who claim that they are Apostles. If you go according to the greek word (apostolos), with the word meaning.."an ambassador; one who is sent;." I can see where a person can claim they are "sent out", but I don't see where anyone in the present day church can fulfill any of the other qualifications of being an apostle.
I realize that others than the 12, such as Barnabas in Acts 14 were referred to Apostles, but even then, it may be possible/probable that he was a witness of the risen Lord.
Additionally, I had an interesting discussion with someone I met yesterday. She indicated that she has a friend, who is a woman, who is an Apostle. I have some serious questions about this. She referenced Junia in Romans 16:7, but from researching this verse, that seems to be a serious and unreasonable stretch of scripture.