For that wattage of Peltier, you are looking at huge heat sink and power fan, unless you are water cooling your Peltier.
The Peltier already says it draws 8A of current. For a normal system config, you need about 4A of current. So you are looking at a power supply that gives 12A at +12volts.
I don't think much 300W PSUs gives that much current. Even my 350W PSU only gives 10A at +12 volts.
Also, since you will be loading your PSU quite a bit, I'd suggest a good name power supply so that you actually get that wattage and current they claim.
For my latest machine (duron700) I bought myself a Enermax 450W PSU (yeh I know it cost a bit) But at least I am assured that I get clean and sufficient electricity flowing through my system.