You should be fine as long as the '1000W' rating is accurate.
I'm using the same board and CPU as you with OC'ed Sapphire 5830's (998/300) with 8GB flash drives on a Corsair 950W going on 2 months now. Not a single issue. 2.5" 7200 RPM HDD's power consumption is negligible. And you're using 4x5850's and 5850's have lower TDP's than 5830's... My rig setup is 620W-640W at the outlet from a Kill-A-Watt. You might barely break 600W with yours.
Cheers,
Kermee