Is this a good gaming pc for under $300?

Solution
Slight change on the memory, and a better quality PSU, only a little more:

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($65.88 @ OutletPC)
Motherboard: Asus H81M-K Micro ATX LGA1150 Motherboard ($48.99 @ SuperBiiz)
Memory: Crucial 8GB (2 x 4GB) DDR3-1600 Memory ($26.99 @ Newegg)
Storage: Hitachi Ultrastar 500GB 3.5" 7200RPM Internal Hard Drive ($37.00 @ Amazon)
Case: Xion XON-310_BK MicroATX Mid Tower Case ($25.99 @ Amazon)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($42.99 @ SuperBiiz)
Operating System: Microsoft Windows 10 Home...

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Slight change on the memory, and a better quality PSU, only a little more:

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($65.88 @ OutletPC)
Motherboard: Asus H81M-K Micro ATX LGA1150 Motherboard ($48.99 @ SuperBiiz)
Memory: Crucial 8GB (2 x 4GB) DDR3-1600 Memory ($26.99 @ Newegg)
Storage: Hitachi Ultrastar 500GB 3.5" 7200RPM Internal Hard Drive ($37.00 @ Amazon)
Case: Xion XON-310_BK MicroATX Mid Tower Case ($25.99 @ Amazon)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($42.99 @ SuperBiiz)
Operating System: Microsoft Windows 10 Home OEM 64-bit ($84.99 @ NCIX US)
Total: $332.83
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-06-06 16:43 EDT-0400

This is about the best you can do for $300ish. Keep in mind that hard drive is still a little small (but way faster now at least) so you may be looking to add more space later, and a GPU like a GTX 750ti would be a good addition. The internal graphics on that CPU are just enough but you may get some slowdowns. The PSU I added can clearly use a GTX 750ti or GTX 950 when you are ready.
 
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AMD APU/SSD option:

PCPartPicker part list / Price breakdown by merchant

CPU: AMD A6-7400K 3.5GHz Dual-Core Processor ($49.98 @ NCIX US)
Motherboard: Biostar Hi-Fi A70U3P Micro ATX FM2+ Motherboard ($41.98 @ Newegg)
Memory: Kingston HyperX Fury Black 8GB (2 x 4GB) DDR3-1866 Memory ($29.99 @ SuperBiiz)
Storage: A-Data Premier SP550 240GB 2.5" Solid State Drive ($54.99 @ NCIX US)
Case: Xion XON-310_BK MicroATX Mid Tower Case ($21.98 @ Newegg)
Power Supply: EVGA 400W ATX Power Supply ($21.98 @ Newegg)
Operating System: Microsoft Windows 10 Home OEM 64-bit ($83.89 @ OutletPC)
Total: $304.79
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-06-06 17:12 EDT-0400