This report analyzes the feasibility of running PlayStation 3 (PS3) emulators, specifically RPCS3, on school-issued Chromebooks. The findings indicate that while technically possible on specific high-end models, the combination of hardware limitations, software restrictions (Google Enterprise enrollment), and legal implications makes PS3 emulation largely unviable for the majority of students using standard school-issued devices.

Is there a simpler way to install RetroArch on a Chromebook?

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The biggest hurdle is the massive performance gap between a PS3 and a typical Chromebook.

at a "playable" level—it is designed for high-end PCs, not lightweight educational devices. Recent breakthroughs have improved performance for low-end CPUs, but this still refers to desktop-class hardware rather than Chromebook chips. Tom's Hardware Safe Alternatives for School Chromebooks

Since your Chromebook probably lacks the power to run the game itself, let a remote server do the work.