Precise underwater positioning remains a fundamental challenge for underwater robotics because global navigation satellite system (GNSS) signals cannot penetrate the sea surface. This paper presents Raspi$^2$USBL, a Raspberry Pi-based passive inverted ultra-short baseline (piUSBL) positioning system that provides a low-cost, accessible, and reproducible platform for underwater robotic research. The system consists of a passive acoustic receiver and an active beacon. The receiver integrates a hydrophone array, multichannel preamplifier, oven-controlled crystal oscillator (OCXO), Raspberry Pi 5, and MCC-series data acquisition (DAQ) board. The beacon integrates a matching network, power amplifier, and transmitting transducer. An open-source C++ framework supports clock synchronization and triggering for one-way travel-time (OWTT) messaging, while performing matched filtering, array beamforming, and adaptive gain control to estimate the time of flight (TOF) and direction of arrival (DOA). The system was validated in an anechoic tank, a freshwater lake, and open-sea trials. Results demonstrate a slant-range accuracy better than 0.1%, a bearing accuracy within 0.1°, and stable performance over distances up to 1.3 km. These findings show that low-cost, system-level reproducible hardware can deliver research-grade underwater positioning accuracy. By releasing the software framework and providing a reproducible hardware architecture, Raspi$^2$USBL offers a reference platform that lowers the entry barrier for underwater robotics laboratories and promotes reproducible research in underwater acoustic navigation and swarm robotics.