Gather data on battery power; how to get it off?

An SD card would be somewhat better as you could read it from a card reader (provided you used FAT on it).

To use the eeprom, you'd have to create a data map and then when connected to the pc, it would send all the stored data.
But depending on the Arduino you have and sampling time, the eeprom might not be enough. :\

Do some math about the number of values you want to keep and the eeprom size.