I won't be able to give a complete answer as I'm not that familiar with the Catholic and Orthodox church structure, but Biblical church structure is something I've been looking into recently so here is what I've found out.
The Pope I believe the argument for having one man overseeing the entire church comes from Jesus statement to Peter in Matthew 16:18 "And I tell you, you are Peter, and on this rock I will build my church ... " (ESV). Since Peter means rock or stone, Jesus' statement is taken to mean that Peter was the foundation of the church and would lead it. He was the first Pope.
The argument against this is that the rock/foundation Jesus was referring to was not Peter himself, but Peter's statement that Jesus was the Messiah in verse 16.
The Cardinals Or any small group of men appointed to oversee the church originates from the concept of the 12 Apostles. I would reference Acts 1:12-26 where the remaining 11 Apostles found it necessary to choose a man "...to take the place in this ministry and apostleship from which Judas turned aside to go to his own place." Also, Acts 15 where "The apostles and the elders were gathered together to consider this matter [of circumcision]." It would seem biblical that a small select group of Church officials meet together to decide how to apply doctrine to life.
Bishops These are comparable to the Elders/Overseers in the Bible. The key passages detailing what an Elder should be like are 1 Timothy 3:1-7 and Titus 1:5-9. Focusing more on church structure, Acts 14:23 indicates that it was important (maybe even necessary) to Paul and Barnabas to appoint Elders in the churches where they went.
Priests Here, we get to the references about Deacons in the Bible. The Greek word simply means 'servant', but considering the stipulations in 1 Timothy 3:8-13 and the selection of 7 men with the laying on of hands in Acts 6:1-7, we can assume that they were appointed to serve in a role that differentiated them somehow from other Christians, though all Christians should be serving.
And so, in the New Testament, we begin to see a structure of Authority loosely take shape. Apostles appointing another Apostle to replace Judas. Apostles sending Paul and Barnabas to plant churches. Paul and Barnabas appointing Elders. Elders (Timothy) encouraged to appoint other Elders and Deacons. Etc. The Catholic and Orthodox church structures use different titles for the different positions of authority, but they have quite a few similarities to the structure seen emerging in the early church.
Lastly, if we take a look back at the Old Testament, we see an example where Moses arranges an authoritarian structure simply to help facilitate the proper workings of a community. In Exodus 18:13-27 Moses heeds the advice of his father-in-law and appoints men "out of all Israel and made them heads over the people, chiefs of thousands, of hundreds, of fifties, and of tens. And they judged the people at all times." This is an indication (not a proof) that even authoritarian structure that is arranged by man is Biblical and "endorsed" by God (supported by Romans 13:1-7, Hebrews 13:17). I won't expand more on that thought here as it goes beyond the scope of the question, but my point is that regardless of the specifics of these church structures, Authoritarian Structure in itself and as a concept is Biblical.
Hope that helps to answer your question. I'm no scholar and I've never been to Seminary, but I guess like C.S.Lewis, even a lay-person can do their own study and pass on what they've learned. I'd love to hear what other people have to say on the topic. Good question.