I used to read any novel I could get my hands on.

Literal

At-random novels [object-を] read [nostalgic-past-ものだ].

手当たり次第に (てあたりしだいに) literally means 'in the order hands touch' — grabbing whatever comes to hand, doing something indiscriminately. ~ものだ expresses nostalgic recollection. Subject entirely omitted.