RTC and EEPROM can share the same I2C bus, and they have their own address (check the datasheet). A0/A1/A2 allows you to put more EEPROM on the same bus, but it won't conflict with the RTC (MSB part of the address is hard-coded in the device, and LSB part can be set up by user depending of the device).

You could also use an SD-card as your storage medium. It is a bit more complicated to use, especially if you want a FAT system (check on this forum there's some examples).

I would go with I2C EEPROM, but storing data in binary rather than ASCII.