Flat-fabrication technology may enable the next generation of gigantic deployable architectures devoted to the detection of faint cosmological signals. We assess the applicability of a multifunctional roll-out structure based on shape memory polymer technology for the realization of a large space observatory to measure the cosmological Dark Ages radio signal. Roll-out solutions offer advantageous properties for probe class missions, such as the capability to morph the shape to achieve sufficient structural performance while ensuring high packaging efficiency. We characterize the feasibility of a roll-out observatory in the context of a 5 years-long heliocentric mission scenario. Our preliminary study demonstrates how a four-250 m-long arms architecture with 150 evenly spaced short dipole antennas potentially meets the basic mission requirements dictated by the Dark Ages science case. We conduct a quasi-static structural analysis considering axial and bending loads acting on the arms to assess the structural properties of the proposed architecture, identifying geometric ranges which enable the structure to withstand expected loads while satisfying mass and size constraints. Printable electronics are considered in the design due to the ease of integration with the polymer substrate. In this regard, we explore two distinct electronics configuration options—centralized and decentralized—discussing their benefits in terms of power demand and data management. If successful, such a design may set the stage for future technological development aiming to realize tomographic measurements of the cosmological Dark Ages.