Picking a hard drive at random and finding a datasheet for it, the Seagate Baracuda appears to have a "working" power draw of approximately 6.5 watts, multiply that by 9 and you get an average power draw of about 63.5 watts. Not much at all.
I would be surprised if that stresses any power supply particularly much.
The thing that will kill your power supply if you're not careful is the startup current of the drives. You see, the problem with spinning platter drives is that they use motors, and motors take a huge amount of current to get them moving compared to just keeping them moving. The Barracuda has a startup current of 2 amps.
That's a nice 18 amps of current required before your drives can spin up to do anything.
Unless you have an intelligent BIOS that spins the drives up one by one you are going to need to make sure that your power supply can put out some rich chunky amps down both the 5 and 12 volt lines (most probably the 12 volt line will get hammered the heaviest) and that's assuming that you don't have something else, like say a processor or graphics card or something trying to use that power at the same time. Assuming the motors get their power exclusively from the 12V line, 18 amps at 12V gives a power draw of 216Watts, a very significant percentage of the full power output of your power supply.
Basically your power supply should have a sticker on it that, along with the "300W" information, should tell you what current (in amps) it can put out on each voltage line. The 5V line is probably going to be something like 5-10 amps, but if the 12V line cannot put out at least 20 amps, then I think you're very quickly going to end up with a power supply that's going to put its feet in the air and do a pretty good Dodo impression.
I'm also ignoring the fact that putting 20 amps down a single set of thin copper wires may not be a good idea in the first place, unless they're pretty chunky wires, and you can actually end up with the wires heating up quite impressively and potentially causing a host of other problems completely unrelated to the matter of just how much power your power supply can put out.
Another thing we're also ignoring here is that the figures that get printed on power supplies are laughable and are (being completely honest) only vaguely related to reality. Check the first part of this Dans Data article on how inflated manufacturer power supply figures are... Chances are the wattage listed on the power supply is a momentary peak wattage and I would not expect the power supply to last very long having to regularly deal with spinning up all those drives.