There is a commercial product which does this: it is costly, but beautifully made (of solid metal) and both safe (in terms of temperature) and silent or near-silent for up to 4 drives. http://www.atechfabrication.com/product ... torage.htm
This cools the drives by conduction which is how drives are supposed to be cooled in a normal PC case (that is why they have metal bodies and substantial screws into a metal cage). Normally conduction leads to noise (in the same way that fear leads to anger and anger leads to hatred) but not with this case because of the internal isolation and the solid exterior casing.
For reasons of cost I have also been thinking about building my own solution. I came up with the idea of hanging the drives vertically off a pair of rails on an individual elastic sling for each drive - the rails and elastic would be easy to assemble, the only slightly part requiring skill would be the supports to go at each end of the rails, but I guess it could be wood or just about any other material. You could put that whole assembly in a plexiglass box if you like, but I can't see the point because the box will then have to be fan ventilated to keep the drives cool, and that will make more noise. Maybe you could use just a piece of solid material between the drives and the room, but open at the back and sides?
You can cool drives just by hanging them in free air (convection cooling). It is best to hang them vertically on the narrow end for best airflow up and down the largest faces of the drive. Convection cooling is helped if you fix metal heatsinks or fins to the drives (especially the sides of the drives) - this was tested on SPCR years ago when people mounted drives on foam pads on the base of their PCs. I'm not aware of any commercial heatsink of exactly the right size for a drive but you can buy small generic heatsinks in any good electronics store, and then I think you could drill holes in those and screw one onto each side mounting hole of the HDD.
It is difficult to avoid cabling for a home-built enclosure making an ugly mess. At least you can simplify power cabling by building your own SATA power cable - crimp-on SATA power connectors are available, for example in Australia: http://www.pccasegear.com.au/prod3336.htm
or in the UK: http://www.kustompcs.co.uk/acatalog/SATA.html
For the data connection, you could look for a pair of SATA multilane adapters with a SATA multilane cable between: http://www.cooldrives.com/sapciadsaian.html
Multilane eSATA (also called Infiniband) gives you a single, neat cable with a firm fastening each end and the drives can be 6 feet or 2 metres from your PC. Because of these adapters, there is no need for a costly multilane SATA controller, you can just use a standard cheap 4-port internal SATA controller.
I'm not convinced that Atech 4-drive case would be that quiet. I'll believe it when I hear it and/or when someone I really trust tells me it is.
But I do think your idea has a lot of merit. That multilane eSATA is used in the Atech device; great solution!
I would build it a bit differently than you suggest. I'd basically make a box to replicate, roughly, the bottom chamber of the P180. The elements needed:
-- old/cheap PSU in back with exhaust going backwards, rigged to start w/o mobo.
-- 4 hdd suspended vertically
-- box has open slot in front, at bottom, with mesh filter and feet ~3/4" high
-- multilane eSATA port at back
The PSU fan would have to be a very quiet one -- set at a fixed speed where its noise is below that of the 4 HDDs. That one fan would be more than enough to cool the drives and itself -- the total load on the system would never be more than ~60-70W (at startup).
Dimensions would be a little wider and taller than a standard ATX PSU and perhaps 14" deep. It could be made out of wood, metal -- whatever you want, really, as long as it's sturdy and not resonant.
If you used 4 very quiet drives -- WD, Samsung -- I think the overall noise could be kept in the low 20s dBA/1m in idle. A bit of panel damping would probably help -- that would make the box a bit bigger, but no big deal if you have it out of the way with a 2m-eSATA cable.
If you wanted to go with different dimenstions -- say lower and wider with two pairs of stacked suspended HDDs -- then you'd have to either hack a standard PSU or, easier, go with a picoPSU.