One reason for using a solid-state disk is that it requires only a 5 V supply. (The other one is that it is silent.) This makes the power supply much simpler than it would be for a PC-style system, such as a mini-ITX design.
The power supply is a 5 V 20 W PCB-mount module made by Powersolve and sold by Wattbits (they also have an eBay store), code PME20-05. They also have bare-PCB and metal-cased versions. The one that I have was a bit awkward because its pins are not on a 0.1" grid, so some hacking was needed to fit it to the veroboard; the metal-cased version (PGE25-05) would have been easier to use.
With the DVD drive idle, power consumption is about 5 W. With the DVD running it rises to about 9 W, and there are peaks of at least 11 W while it spins up. Powersolve have lower-rated supplies than the one that I'm using but I think 15 W is as low as you could go.
I guess that many people would probably choose to use an external power supply brick or wall-wart. (So far I have managed to not electrocute myself on this, but care is needed.) If you do that, I suggest that you fit a capacitor at the input to the case; the iMX53 board has very little input capacitance, and the inrush current when other devices turn on can cause it to fail especially when there is significant inductance in the power supply lead. You might even want to use a 12 V external supply and fit a DC-DC converter in the case.
I decided to use a trustworthy powered USB hub - this one is a D-Link - partly because of these concerns about power. Unlike many hubs, this one has per-port power control.