The data could be shoved into the EEPROM of the Arduino itself. Since the ADXL330 is ten bits per channel, you will need at least four bytes per sample to store them at full resolution (four bytes is 32 bits). If you sample ten times a second, then an ATmega328 could store about 25 seconds worth of data.
If you sample more often, memory will fill up faster. If you need to sample a longer timespan, memory will fill up. There are external EEPROM packages which are quite easy to use with minimal support. They come in various capacities but are still in the kilobytes range.
There is also an SD card option (like Adafruit's WaveShield), but the programming is fairly complicated to get into. The major wins for that approach is that it becomes super-easy to get huge data onto the PC: eject card, insert card into PC, browse one or more data files.
Lastly, before you go too far with logging acceleration data: you won't get anywhere near the time or accuracy required to be able to "dead-reckon" or figure out the position of the object from its acceleration profile. You'll get a rough estimate, at best. These are noisy little sensors.