Cool
Are you set on the 1RU format for some reason? I’d imagine other size options might be more space-efficient but if you just have 1RU to play with that’s fair enough…
If you are buying new / wanting reliability / money is no object etc then some sources are noted below, but if you want to build it as cheap as possible I’d suggest getting along to the BuyEquip market day that’s on the 24th, pick up an old 1RU server for $70 or less and use the case and PSU(s) in that to build your system around. You might be able to repurpose drive mounts in the existing case as well, though you’ll probably end up doing a fair bit of metalwork either way. You’d also be able to pick up a switch and probably some other bits and pieces as well, very cheaply.
If you’re going new… I’d check out U-mart for switches and other general gear.
For the PSU either secondhand 1RU server PSU’s (which you’ll have to modify to make them turn on - there is info out there on how to do that as a lot of people use them for charging LiPO batteries) or you could try non-computer-specific ones like the two Jaycar ones below. You might need to mix and match to get the rails you need or they might have a model that does what you want.
https://www.jaycar.com.au/medias/sys_master/images/8970584227870/mean-well-320w-12v-26-7a-power-supplyImageMain-515.jpg
320W 12v 26.7Amps $169
https://www.jaycar.com.au/medias/sys_master/images/8970584752158/mean-well-150w-12v-12-5a-power-supplyImageMain-515.jpg
150W 12v 12.5A $69.95
“Small period of time” is a relative concept. A huge bank of capacitors might be the best bet, where again, “huge” is relative. If milliseconds, then you might not need anything depending on the PSU quality, or a handful of caps might do it, depending on load. If you have specifics on what is going to cause the spike in load some here will be able to make recommendations, I’m sure. If it’s for sustained periods then speccing your PSU higher might be the best answer. Adding a battery means you then have to take care of charging it and maintaining it, and monitoring when it dies so you know it needs replacing (Valar Morghulis, and all batteries must die also).
What needs all the power? A Pi3 apparently draws 3.7W when pegging the CPU, 24 pi’s gives a bit under 90W which is where I guess your 100w 5v figure comes from, but another 100w or so for 2xHDD, switch and NUC? Not saying it’s wrong, just wondering what the specs are.
If your case is going into a populated rack cabinet in a datacentre, I’d consider putting the fans on the front or back, drawing air in from the front and exhaust to the back - which is how most rack-mount IT gear seems to be set up. If it’s going to live at home then use whatever gives you the appearance and noise levels you desire 
Sounds like a cool project, are you able to say what it’s for or does the NSA not need to know about it?