Dead ReadyNAS...how to recover the data

Given the completely outrageous prices on those replacements, you would need to have some very valuable data before I'd recommend getting one.  You can find much more reasonable prices on used ones on eBay.  While you'd have no guarantee of how much life you might get from it, you'd likely be safe for long enough to recover the data.

Another option is a temporary or permanent power supply replacement.  For simple recovery, the temporary option is probably your best plan.  While the ReadyNAS connection looks like a standard 20-pin ATX one, it is slightly modified.  The -5V and -12V connections are replaced with additional +12V to give more power for the drives.  A diagram is here: ReadyNAS_PSU_pinout.  It is pretty easy to take a 24 to 20 pin adapter/extender for an ATX power supply and wire it to do the swap.  Then use it externally with a standard supply long enough to recover the data.

Alternately, replacement supplies are available (though also pricy), and a standard Flex ATX supply can be made to work internally, needing the pin swap plus doing something about the fan.