Old ReadyNAS comes back from the dead - Very weird

Bad capacitors can heal themselves (an internal short finally blows open like a fuse), though it doesn't usually last for very long.  The most likely place is the power supply.  The fact that it started on it's own may point to the 5VSB (5 volt standby) output of the power supply, which powers the on/off circuit and is also always on unless the switch on the back is off or the unit is unplugged.  But the unit will also come on on it's own if mains power is cycled, so it could be something else in the supply.

As for a replacement, you need an SFX format one.  Though the original is 300W, I recommend at least 350 for the replacement since the original has more 12V current than most 300W.  Higher current is also fine, though adds to cost.  You are going to need an extender for the 24-pin connector, which is a standard PC ATX one.  Most need about 5", but you may want to wait till you get the supply to measure yourself.  You'll also need three 4-pin Molex connectors (old-style hard drive power connectors).  Most supplies made today don't have that, so you'll need at least one Y cable and/or SATA to Molex adapter unless your replacement supply is cabled such that you can move the 90° Molex connectors from the old to the new (which I like to do, because I can also trim the cable length).

There is an eBay vendor who sells SFX suppies already modified for the 6-bay legacy ReadyNAS if you want to take the easy (and not that much more expensive) way.  I've not tried them, and have no affiliation, but others have reported success with them.