Yes, of course this is possible.
The SD card, whether it is micro SD or Mini SD, or Standard SD, can communicate with your microcontroller via the SPI protocol.
There is a library available from HiTech which will allow you to use the SPI protocol with ease.
It can work at any low speed such as from your 4 MHz crystal, but please remember that there is a maximum speed, although you would be unlikely to achieve it with a regular microcontroller.
PLEASE do remember that the SD card operates at a voltage of about 3.3V, but your microcontroller operates at about 5V, so you must use a level shifter in order to not burn out the SD card due to the excessive voltage.
There are MANY tutorials available on the web about this exact subject.
Good luck!