300/8 = 38 shift registers, with a current limit resistor needed per LED.
If you use MAX7219 instead, each controlling 64 LEDs, then only 5 chips are needed.
Each MAX7219 is essentially 8 shift registers, and you will only need one current setting resistor per chip, saving you a lot of components.
You will need a 10uF, a 0.1uF, and a 10K resistor per chip.
I offer a board that brings out a pair of wires per LED if that will help with wiring:
This shows it with female headers, most folks ask for male headers instead and you can use 30 AWG wire wrap wire from the males pins to the LED legs as I did here:
