This isn't just the best turntable under $300, it's the single most recommended record player for just about anyone who seeks my advice on the subject. Once you get to the price range of $300 for a record player, you are starting to get to models that will last many years beyond the cheaper and lesser models available out there.
Audio Technica AT-LP120BK-USB turntable (typically $295)
The AT-LP120BK-USB is hands down my number one most recommended turntable and it's the best record player under $300 with absolute certainty. It is fully upgradable, plays records with precision at 33 1/3 RPM, 45 RPM as well as the sometimes overlooked 78 RPM, it has built-in phone pre-amp so you can plug it directly into power speakers or is equally well equipped to plug into a receiver, it has pitch adjustment, anti-skate, adjustable counter-weight so you don't wear out your records unnecessarily, height adjustment, and essentially every option you need for a high-quality record player set up. The choices are unlimited with this model. Yes, you could spend more for turntables that will do the same at a higher price, you could spend less and get turntables that aren't this quality and don't have all the same features which could damage your records or make your stereo sound bad, but this model is at the crossroads of quality versus price on the turntable spectrum. If you want to buy a turntable, look no further.
The Best Turntable with Speakers under $300
Teac LP-R550USB (typically $280)
I think the upper limit of just about any all in one turntable or record player with built in speakers is naturally under $300, so this model represents one of the most expensive all in one systems you can buy and still comes in less than $300 total. I don't recommend because I think the primary option above is the best one you can buy, but if you're looking for a record player with a lot of extras, then maybe this is the one for you. You can also see the lineup of all-in-one record players on this site over HERE if you are interested.