The regions around the celestial poles offer the ability to find and characterize long-term variables from ground-based observatories. We used multi-year Evryscope data to search for high-amplitude (~=5% or greater) variable objects among 160000 bright stars (m_v_<14.5) near the South Celestial Pole. We developed a machine-learning-based spectral classifier to identify eclipse and transit candidates with M-dwarf or K-dwarf host stars, and potential low-mass secondary stars or gas-giant planets. The large amplitude transit signals from low-mass companions of smaller dwarf host stars lessens the photometric precision and systematics removal requirements necessary for detection, and increases the discoveries from long-term observations with modest light-curve precision among the faintest stars in the survey. The Evryscope is a robotic telescope array that observes the Southern sky continuously at 2-minute cadence, searching for stellar variability, transients, transits around exotic stars and other observationally challenging astrophysical variables. The multi-year photometric stability is better than 1% for bright stars in uncrowded regions, with a 3{sigma} limiting magnitude of g=16 in dark time. In this study, covering all stars 9<m_v_<14.5, in declinations -75{deg} to -90{deg}, and searching for high-amplitude variability, we recover 346 known variables and discover 303 new variables, including 168 eclipsing binaries. We characterize the discoveries and provide the amplitudes, periods, and variability type. A 1.7R_J_ planet candidate with a late K-dwarf primary was found and the transit signal was verified with the PROMPT telescope network. Further follow-up revealed this object to be a likely grazing eclipsing binary system with nearly identical primary and secondary K5 stars. Radial-velocity measurements from the Goodman Spectrograph on the 4.1 meter SOAR telescope of the likely lowest-mass targets reveal that six of the eclipsing binary discoveries are low-mass (0.06-0.37M_{sun}_) secondaries with K-dwarf primaries, strong candidates for precision mass-radius measurements.