I am surely no expert, but I can start with some of the preliminaries-
Do you have a source, or do you need a whole chain for $300?
With the budget constraint, how do you feel about used gear? You’ll likely get MUCH more bang for your buck with used gear than new.
A space that small could easily be filled with a decent pair of bookshelf speakers and a simple stereo receiver. If you already have a desktop DAC with RCA outs, powered monitors could also be a decent option. However, the lower cost budget powered speakers (such as sold for computer use) generally don’t hold up to either passive bookshelf speakers or powered studio monitors, and tend to be disappointing in sound quality.
If you need the whole chain to fit into $300 and you’re after good sound, you’ll likely need to scour craigslist. Look for a speaker that is front ported or sealed, since it can go closer to a wall without ruining how it sounds so much (rear ported speakers really should be placed a good foot plus from a wall behind them). You shouldn’t have any trouble finding units from JBL, Acoustic Research, Cambridge Soundworks, Boston Acoustics, or (if you’re lucky) Advent. I see offerings as low as $40 for a pair for speakers that get pretty good marks over at AudioKarma (for their price - they’re not going to compete with brand new $500 speakers). If you need new, and inexpensive Sony SSCS5’s seem to get a lot of love from reviewers for speakers in that price range.
A stereo receiver from the early to mid '90’s can sound fantastic and can be had for a song. I am partial to Yamaha for the variable loudness control which helps tame the mid-forward sound when playing at lower volume, but any of the major brands (Sony, Panasonic, Harmon-Kardon, etc.) had offerings in that timeframe that sound way better than their current price would indicate - the cheaper plastic construction and fairly ugly “black box” look to the early '90’s gear is a real turn off for many, but for “sound quality returned per dollar spent”, these are hard to beat.
If you want to hook up a receiver to your computer to play sound from your computer on your stereo speakers, you will need a way to send the audio signal to the receiver. This is most simply solved by getting a separate DAC. It doesn’t have to be anything fancy, and SMSL M100, or Schiit Modi, or Topping E30, there’s no lack of $100 DACs that will do the job.
The chain would go like this: computer USB port → DAC → stereo receiver → speakers.
This would also let you expand, customize, or upgrade later. Want a CD player? No problem, plugs right in to the receiver. Tape deck? Same. Play music from your phone instead of computer? Shouldn’t be an issue, just plug it into the DAC USB in. Later on, want a more powerful amp? Integrated amps and upgraded receivers aren’t hard to find, though they can certainly get costly. Better speakers in a couple of years? No problem, just replace the speakers and keep everything else, and so forth.