I've seen dozens of I2C eeproms, but have no idea what a 24C0256 is. [why can't anyone
ever have something standard?]
http://www.digikey.com/product-search/en/integrated-circuits-ics/memory/2556980?k=24c02
Note - as mentioned earlier, 24C02 and 24C256 have very different addressing schemes.
First off, you really really need to know what you really have.
Also, you really really need the pullup Rs with I2C devices, despite what may "seem" to
work. As I understand it, the I2C pins on the standard Arduino chips are both open-drain,
so I've no idea why your chip should be working without the pullups.