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This paper introduces a novel approach to the co-design of sustainable embedded systems through multi-objective design space exploration (DSE). We propose a two-phase methodology that optimizes both the multiprocessor system-on-chip (MPSoC) architecture and application mappings, considering sustainability, reliability, performance, and cost as optimization objectives of equal importance. Unlike existing approaches, our method thereby accounts for both operational and embodied emissions, providing a more comprehensive assessment of sustainability. The first phase employs intra-application DSEs to explore Pareto-optimal constraint graphs for each application. The second phase, an inter-application DSE, combines these results to explore sustainable target architectures and corresponding application mappings. Our approach incorporates detailed models for embodied emissions (scope 1 and scope 2), operational emissions, reliability, performance, and cost. The evaluation demonstrates that our sustainability-aware DSE is able to explore design spaces overlooked by traditional approaches, supported by superior results in four key objectives. This enables the development of more environmentally friendly embedded systems while still achieving high performance and reliability.