There's no computer fast enough to run all the roms, so I'll assume you mean most of them. For a completely silent solution, there's a huge case by Thermalright that uses only heatpipes. But it costs something like $1000, so that's kind of extreme.
Those points aside, my recommendations are:
Try to get a "silent" case, like the Antec P150 (aka SOLO I think). It's made from thick steel with some kind of rubber padding that cuts down on noise. They're heavy, but muffle the noise.
Avoid using any fans smaller than 120mm. They make a *lot* of noise. I mentioned the Antec P150 - I should point out it only takes 80mm fans on the front. That's not so good.
Try to get a motherboard that uses heatpipes instead of fans for the chipset cooling. I know some Gigabyte motherboards do, I'm sure there are plenty of others. Check the product photos on manufacturer web sites.
Aim for a Core 2 Duo CPU. They use less power and generate less heat, so there's less fan noise. I guess the low end CPUs like the E1200 should be the quietest. If you need more power, the E8200 uses a 45nm design and runs pretty cool. Underclock your CPU if you like.
For your graphics card, if you're not going with onboard, get one with passive cooling. I'm using a GeForce 8500 and it doesn't even need a fan. Just make sure there's case airflow going past it.
For hard drives, fewer is better. Western Digital have released a 640GB drive that only has 2 platters, which should mean less noise. I believe there's a single-platter 320GB drive available or available soon.
Make sure you have good cable management in your case, and set your airflow to run from the lower front towards the upper back (in general). Connect any case fans and PSU fans to motherboard headers, so the heat sensors can run them slower when possible.
I've found many retail processors with heatsinks don't have the thermal paste properly applied. So if you know what you're doing (or know someone who does), then remove them and reapply your own thermal paste. Better heat spreading means better cooling, so the fans don't have to spin as fast.
There are a few other things you can do like mount your hard drives on rubber grommets, but I personally can't tell the difference.
If that all sounds like too much work, see if you can get hold of one of those slim form factor office computers (Dell and HP make some, I think). They run pretty quiet.
Don't be afraid to use low-end name-brand components. They're often slower, and therefore cooler and quieter.