Arduino sketches are bigger than 2k. The serial bootloader alone is 1k. You could use an ICSP instead of the bootloader and save that space.
The simple "blink a single led on and off" example takes 3676 bytes in Arduino-0007 so that is out. The in-development Arduino-0008 code blinks an LED in 876 bytes, so that might be possible, but any libraries would probably put you over the top. You won't be able to use the serial port library, for example.
RAM space is also going to be a problem.