I think that museums became a little bit unfashionable as a way of knowing or showing something.

unfashionable 是由前缀 un-（表示"不、非"）和形容词 fashionable（时髦的）组成的形容词。fashionable 来自名词 fashion（时尚）加上后缀 -able（表示"能够...的"）。unfashionable 合起来表示"不时髦的、过时的"。