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That’s going to be pretty tough to do, with decent quality, for $300, unless you DIY.
Parts Express may be your answer.
They sell subwoofer kits, with very high quality woofers, and CNC "flat pack" enclosures. No real wood working skills needed, just some basic gluing, and finishing.
Here is a 12" kit for $218:
Then add a plate amp, with adjustable crossover,.like this one for $90:
So for just a bit over $300, you will end up with something that will pretty much stomp all over the sound quality of anything commercially available from a "name brand".
So, unless you want to brag about owning some name brand, don't overlook DIY. Especially if you want better SOUND quality.
Here's 7 other options for $300:
Pay attention to how deep each sub is rated at meaning how deep, measured in Hz, the sub can play. Most recorded music contains bass as deep as 20 Hz which is as deep a bass tone as a typical human can hear.