A new thermodynamically stable, aperiodic "bricks-and-mortar" (B&M) cellular mesophase structure is reported in PS1-b-(PI-b-PS2)(3) miktoarm copolymer and PS homopolymer blends [PSI, long polystyrene; PI, poly(isoprene); PS2, short polystyrene], where PS comprises discrete hard "bricks" and PI the continuous soft "mortar". The mesophase is unique in its extreme domain volume fractions, its lack of positional order, and quasi-long-range orientational order. On the basis of this unusual mesophase structure, a series of PS-based thermoplastic elastomers are realized, combining rigidity from an exceptionally high content of discrete glassy PS domains (up to 82 wt %) and high extensibility with recoverable elasticity from a low content of continuous rubbery PI (down to 18 wt %). The new elastomers show sharp yielding behavior while maintaining good elasticity at large strains. Tensile-SAXS experiments reveal that voiding plays an important role for the mechanical behavior and voids can open/dose reversibly with/without loading. Plastic deformation only results in a slight loss of recoverable elasticity.