Final answer:
The output of a eukaryotic cell system is ATP, which is generated through the metabolism of glucose. The ultimate energy source for this process is the sun, as it powers photosynthesis, which produces glucose.
Step-by-step explanation:
The output of a eukaryotic cell system is adenosine triphosphate (ATP), which is the primary energy carrier in cells. The system requires an input of glucose to function, as it is the primary fuel for cellular respiration, during which ATP is produced. Ultimately, the energy to produce glucose and thereby ATP comes from the sun, as photosynthesis in plants converts solar energy into chemical energy stored in glucose, which is then available to other organisms through the food chain.