...not possible on that budget, but otherwise (SSD & quiet factors) all roads lead to the MacBook Air.
Or, if the Air isn't your style, a MacBook Pro with an SSD. I don't know if you can get those standard from Apple, but it's a great combination, and usually literally silent.
Sure that works just fine for noise and performance but that 700$ budget will be blown to smithereens.
Aye. And for a desktop machine, get a passive-dissipation watercooling solution.
Though there's no way you'll build the box for $700, you'll eat half the budget with a good case and a passive watercooling.