We introduce the \emph{Private Structured-Subset Retrieval (PSSR)} problem, where a user retrieves $D$ messages from a database of $K$ messages replicated across $N$ non-colluding servers, and the demand is restricted to a known structured family of $D$-subsets. This formulation generalizes Multi-message Private Information Retrieval (MPIR) and captures settings where the demand space is constrained by application-specific structure. Focusing on balanced ${\{0,1\}}$-linear schemes, a class that includes several best-known MPIR schemes, we derive converse bounds on the maximum retrieval rate and minimum subpacketization level required to achieve any given rate. We also develop an optimization-based framework to construct schemes for general structured demand families, providing flexibility in optimizing the retrieval rate or the subpacketization level. When specialized to the full demand family, this framework recovers known balanced $\{0,1\}$-linear MPIR constructions; for more restricted demand families, it can exploit the demand structure to increase the retrieval rate, reduce the subpacketization level, or both. We demonstrate this through a structured-demand example in which the proposed PSSR scheme simultaneously achieves a higher rate and requires a smaller subpacketization than the best-known MPIR scheme for the same parameters $N$, $K$, and $D$. Our parallel work on contiguous-demand families further illustrates the scope of this framework by yielding rate-optimal schemes with substantially smaller subpacketization and no field-size restrictions, improving upon MPIR-based schemes.