Here is what I would do with that budget:
http://pcpartpicker.com/p/GsfVgs
PCPartPicker part list:
http://pcpartpicker.com/p/GsfVgs
Price breakdown by merchant: http://pcpartpicker.com/p/GsfVgs/by_merchant/
CPU: Intel Core i3-4150 3.5GHz Dual-Core Processor ($99.99 @ SuperBiiz)
Motherboard: ASRock B85M-ITX Mini ITX LGA1150 Motherboard ($67.98 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($59.99 @ Newegg)
Storage: Crucial MX100 128GB 2.5" Solid State Drive ($59.95 @ SuperBiiz)
Case: Cooler Master Elite 130 Mini ITX Tower Case ($31.99 @ Micro Center)
Power Supply: SeaSonic 350W 80+ Bronze Certified ATX Power Supply ($33.99 @ SuperBiiz)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($14.99 @ SuperBiiz)
Total: $368.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-02-10 04:00 EST-0500
It is about as bare bones as can be and still have an i3, 120gb ssd, and 8gb of ram. No operating system, not sure if you were figuring that or not.
You could get a little cheaper with a crappy case and power supply, but wouldn't recommend it. : )