£300 will easily get a quality 2nd hand amp and decent 2nd hand or new bookshelf hi-fi speakers, along with some stands if you need them (depending on how the speakers sound sat on a PC desk - probably not too great) and cables.
If I wire my PC to my hi-fi (Marantz PM66-SE, Marantz CD5000, Eltax Monitor 3, Atacama SE24) games sound awesome. That lot cost me under £250 from ebay, less without the CDP which you wouldn't need.
Just make sure to get shielded speakers if you have a CRT.