Unless you have alot of spare parts from a previous pc you are asking the impossible, an operating system license alone will set you back 1/3 rd of your budget. Isn't there a government programme where you can get a cut price/free computer? Either way you will never ever get anything even close to 'good for gaming' for $300.
Try employment, you will quadruple your budget in a month even working at McD's and then we can talk about good for gaming.