Context: Tadano 只野 thinks about the school's most popular girl.
Komi-san wa gakkou no madonna demo hanashi-kake-dzuraibijin demo nakatta. 古見さんは学校のマドンナでも話しかけづらい美人でもなかった。
Komi-san was neither the school's madonna
nor a beautiful person [who's] hard to [approach].
hanashi-kakeru 話しかける
To approach and talk with. To start talking to. To begin a conversation with.
Very literally: to pour a conversation onto someone.
In this panel, Tadano says he thought Komi-san was unapproachable because she was a "beautiful person." In other words, it's ~dzurai because she's way out of his league. This is a subjective matter. Someone at the same league as her wouldn't face the same difficulties.
