Michal,
... low power when running, low power when sleeping...?. Most 8051 have a quiescent current of microamps when in power-down mode, but they need more than 10 mA when executing code. I don´t know any 8051 device with less than 1 mA in operative mode, though this current depends on the crystal frequency yo use.
For battery operated applications (for example a wireless PIR detector), designers usually keep the micro in power-down mode until some external event takes place. In order to leave the power-down mode, the micro has to be reset. Some devices can wake up via an external interrupt.
I have seen some applications where the MCU is reset cyclically (say every minute), it performs some operations and then enters power-down mode again. A low power 4000 series CMOS oscillator/counter can be used to generate the reset pulses.

Regards,
Alfredo del Rio.