Saying the Rosary and repeating the Hail Mary 50 times seems strange and irrational to me. Are we supposed to concentrate on the prayer we are saying and the mystery all at one time?

The Catechism speaks of the rosary as an epitome of the whole Gospel (CCC # 971). In other words it is a summary or an embodiment of the whole Gospel, because it encourages the faithful to meditate upon the fundamental events of salvation history, and the life of Christ. Someone has called the rosary, the “gospel on the string.”
 
These images help to emphasize that the central work of the Rosary is to meditate on the Scriptures and that Mary’s work, above all, is to lead us to a deeper faith in Jesus.
 
There are some who find the Rosary a helpful way to pray. Others find it difficult and distracting. Thus the Church, while encouraging the Rosary, does not require it. It is a fact that people are suited to different prayer forms. Saying repeated Hail Marys, is appreciated by many as a kind of rhythmic background for the meditation on the gospel passages, which is the key point of the Rosary, and this aspect of it should be central in your thoughts.