The memory wall is a major bottleneck for continuing to improve the energy efficiency of computing systems. To overcome this challenge, various nanomaterials, devices, circuits, architectures, and three-dimensional (3D) integration techniques are under development for future memory solutions. However, major trade-offs exist when designing memories to achieve high on-chip memory capacity, high retention time, high endurance, low access times, low access energy, and low static leakage power. We present an energy- and area-efficient embedded DRAM memory architecture (quantified by EADP: the product of total energy consumption, circuit area footprint and application execution time) that leverages monolithic threedimensional (3D) integration of three types of field-effect transistors (FETs): (i) Indium Gallium Zinc Oxide (IGZO) FETs for ultra-low off-state leakage currents enabling high retention time DRAM; (ii) Carbon Nanotube FETs (CNFETs) for high on-state drive currents leading to fast access times; and (iii) Silicon CMOS for its combined energy efficiency and low off-state leakage current (for memory peripheral circuits implemented on the bottom physical circuit layer). Our resulting 333-eDRAM achieves each of the following simultaneously, which we quantify and describe how to co-optimize in this paper: high density, high retention time, high endurance, low access times, low access energy, and low static leakage power. We show full physical layout designs detailing how to implement 333-eDRAM and quantify EADP for an ARM Cortex-M0 processor + on-chip 333-eDRAM implemented at a 7 nm technology node, running applications from the Embench benchmark suite. Using cycleaccurate simulations of applications, SPICE circuit simulations, compact models calibrated to experimental data, and detailed full physical layout designs of 333-eDRAM memories, we show that on average (across 16 Embench benchmarks), ARM CortexM0 + IGZO/CNT/Si 333-eDRAM offers $1.96 \times$ better EDP and $5.15 \times$ better EADP than ARM Cortex-M0 + Silicon eDRAM.