Do you already have the OS? If not, that'll be $80-$100 off your budget.
It's REALLY hard to make a gaming pc for ~$300.
That build you linked needed a lot of work, and at that budget I wouldn't get a CPU/GPU, but rather an APU (can add in a GPU later once you get the money).
Here's what I've made if you're willing to spend a little over $300.
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: AMD A10-5800K 3.8GHz Quad-Core Processor ($104.29 @ Amazon)
Motherboard: Gigabyte GA-F2A55M-HD2 Micro ATX FM2 Motherboard ($42.99 @ Newegg)
Memory: G.Skill Ripjaws Series 8GB (2 x 4GB) DDR3-1600 Memory ($49.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Case: Rosewill FBM-01 MicroATX Mini Tower Case ($22.49 @ Amazon)
Power Supply: Corsair Builder 430W 80+ Bronze Certified ATX Power Supply ($24.99 @ Newegg)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer ($14.99 @ Newegg)
Total: $319.72
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-07 17:40 EST-0500)
You can probably play the newer games at around medium settings.
Definitely can max out the valve games.
That APU is the best you can get for that price.
Edit: You shouldn't even consider an SSD at that budget.
Din625's builds have 32gb of storage, that's only enough for your OS alone!
Don't even bother, just get a WD Blue it'll do just fine.