It is possible to use 74HC595 for a LED cube.
A matrix is 2 dimentions = X and Y So a cude need 3 dimentions = X , Y , Z So one chip controlling X, one chip controlling Y and one chip controlling Z. The numbers of LEDs is : 8 * 8 * 8 = 512 Also, because of the currents per leds, you will need a transistor to control the leds axis with enough current rating ( 8 + 8 + 8 transistors) , limiting resistors for each leds ( 512 resistors ).
So you will need to build just one cell and repeat the cell. To see what I am trying to say, just reads about Leds Matrix, 7 segments displays ( commun anode or cathode ) to get some ideas how to connect a 3 dimentional leds cube.
That my help at this time. I did not build a led cube, YET