There are no 1GB SSDs. The smallest you will find are probably 16GB and generally the smaller an SSD is the slower.
There are 1GB USB flash drives and 1GB SD cards you could connect to the computer but stilll And 1GB is not enough for XP to sit on.
Without knowing what this computer is to be used for there is no way to make good recommendations.
Fans are pretty much necessary to keep a computer from overheating. Depending on the CPU you can get a fan-less CPU cooler and have everything else fan-less but intake/exhaust fans are more or less irreplaceable. Without them the case would heat up and the whole computer could overheat. You can get inaudible fans to fix the cooling problem without making noise.
The larger a fan is the slower it has to rotate to move x amount of air and the slower it rotates the less noise it makes so if you get large fans (120mm and larger) then you shouldn't hear them. Some cases support very large fans like 200mm, 220mm, or 240mm and I've even seen 360mm before and I can pretty much guarantee that you won't hear them.
To further help the noise you can get a PSU with a large fan.