Your quality options in that price range will probably by limited to well used Remington 1100s, older Remington automatics, and the Winchester 1400.
The Remington 1100 was the first highly successful gas operated shotgun, and it is pretty durable. If you find one in your price range you should check the rear of the bolt handle slot in the receiver for cracks, and change the O-Ring. Some replacement of springs might be in order too. The 1100 is availiable in 2.75in and 3in configurations; the 3in guns will generally not cycle 2.75in loads. The 11-87 is basically an 1100 variant that will cycle both 2.75in and 3in shells without adjustment.
I would stay away from many of the older Remington autos, they tended to not have long production lives and it can be hard to find spare parts. One exception is the Remington Model 11; this gun is a licensed copy of the Browning Auto-5. They are hell for stout and recoil operated. However, they are 2.75in only and need to have the recoil setting changed based on whether you are shooting a light or a heavy load. There is also the 11-48, which is basically a redesign of the Model 11 to make it more cost effective to produce. These can be finicky as Remington tried to eliminate the need for different recoil settings, not totally successfully. As a rule of thumb if an 11-48 won't cycle the magazine tube needs to be oiled more.
Lastly, the Winchester 1400. These guys are real easy to find under $300. They are gas operated and 2.75in only, and are limited to three shells. The gas system is mounted in front of the magazine, so there is no way to extend the capacity.
There are other good models that you might find on a budget of $300, but there is also a lot of crap too. Unlike pumps there is a lot that can go wrong with a semi, so you really need to check a prospective purchase.
EDIT: None of these in their most common configurations are set up with short barrels. If you just want a defensive shotgun get a pump.