Prius beeps after being turned off, then won't start

Another post that illustrates the futility of going to the Toyota dealer and asking for your 12V battery to be checked. Instead, it makes more sense to apply that $69 (multiplied by a few times) towards buying a new battery.

Your current 12V battery (which contains six cells within) has a shorted cell. This is why the voltage measures 10.3V. Each cell, when charged, will produce ~2.1V. Hence your battery has five cells that are operational.

You can wait until the battery dies, if you are comfortable with the idea that the car may be inoperative at an inconvenient time. Otherwise it would be good to replace the 12V battery at four-year intervals if you want to improve the car's reliability.