It is very simple to turn the ATMega328, used in the standard Arduino, into an RTC.
Replace the crystal with a 32 kHz watch crystal, which acts as a clock source for Timer2 (asynchronous mode) and use the internal 8 MHz oscillator for the rest of the CPU.
Described here and elsewhere on the web.
It won't be as accurate as the DS3231 as the crystal is not temperature compensated, but the RTC can keep running while the rest of the processor sleeps, so it is quite useful.
If you are interested I can post Arduino code to implement the RTC.