Wikipedia takes a similar phrase as an example under mediopassive voice
A few examples of unaccusative verbs in English with meanings similar to a mediopassive:
- The book reads well.
- The trousers wash easily.
- Ripe oranges peel well.
- The book was not selling.
Further up it gives a short historic overview, importantly telling us:
A number of Indo-European languages have developed a new middle or mediopassive voice. Often this derives from a periphrastic form involving the active verb combined with a reflexive pronoun. This development happened independently in the Romance languages, the Slavic languages, and the North Germanic (Scandinavian) languages.
Indeed, German, though not strictly North Germanic, would also say "Das Buch ließt sich gut" ("The book reads [sich = reflexive pronoun] well"). Suffice to say that there is good reason to believe this construct was inherited into the 1700s (but whence came it?); an Old English equivalent shouldn't be too hard to find, if it existed itself. Answers and readers that don't like each other very well are unfortunate, but require just a little willingness to compromise.
@AndrewLeach's notion, interpreting the phrase as "books that [I] never read" does not explicitly suppose ellipsis, but a semantic surface analysis that would be equivalent with "books that [were] never read [by me]". Andrew argues for 'distant modifiers' and the point is well taken, because it's needed to explain the adverb of time (a book doesn't have time, people do). I argued for the passive reading, because an apparent ellision of the auxiliary passivising verb is whitnessed in archaic literary German texts (actually more likely if not exclusively in active perfect constructions). Neither argument, basically implying ellipsis, is fully convincing. The apparent ellipsis is the reason that the phrase sounds wrong. It might be explained by a variant of mediopassive that became derilict perhaps via ellision (more likely at the end of a sentence) or morpheme loss and hapology (sound change).
PS: I see only now that Andrew had meant approximately "what should I, that [= who] never read for half an hour, do with books". This does seem not too alien, and would be obvious from verbal emphasis. My variant would need a set phrase that derives from "the book reads well". I'm not aware of such development, nor of "that" used as a personal relative pronoun refering to "I". Effectively, this answer is a lot of hot air.
A detailed explanation would belong on linguistics.SE. The uses of the mediopassive voice are diverse, so it's not clear in which way this phrase should be accepted as correct; From the descriptive point of view it is enough to acknowledge that it is existant and understood. A German might as well ask why a book should read itself (indeed, that's a misleading translation of the pronoun in this case).