As you can see, you opened a can of worms with this question. I own all of the rifles that have been recommended so far, with the exception of an Enfield No4, but I do own an Enfield No1 and an Ishapore 2A.
Enfields are wonderful. I love both of my Enfield-type rifles. But if you're looking for the "best" buy under $300, then I guess you have to think about what you want the rifle for.
The Enfield may very well be the "best" if you want a collector's piece with rich historical value. They're rugged and proven on the battlefield, but many of them have seen decades of hard use and less than loving storage. They're built a little loose intentionally, which goes a long way toward maintaining reliability in mud and dirt, and their bolts function quickly and smoothly with a little TLC. You may hear people warning you about the problem with excessive headspace on Enfields, but the good news is that it's easily corrected simply by replacing the bolt face. On some other military bolt actions, you'd have to rebarrel the weapon. Generally, the No4 rifles are in better condition than the No1 rifles, but that isn't always the case. If you buy a No5 "Jungle Carbine," be aware that it may not be a real No5. It could be a converted No1 or No4.
If you're talking about the best bang for your buck in a rifle that you're going to frequently shoot, I'd go with a Mosin-Nagant of some sort. M38s, M44s, and 91/30s are all easily available for well under $100 in very good condition. These rifles are simple and rugged, and the design was proven on the battle field in the Winter War, World War II, and other conflicts. These rifles were produced in very high numbers, so they may not be as "collectible" as Enfields yet, though some, like the Finnish models, fetch higher prices than others.
As for the K31, it could be the "best" if you're looking for a finely built, interesting, accurate rifle. The straight-pull bolt, properly cleaned and greased, is slick and a joy to operate. Accuracy is amazing, especially considering the price of the rifle. These weapons have the advantage of never having been used in front-line combat. They were carried under harsh conditions, but not fired as much as other military rifles their age, which is why the metal is usually in fantastic condition while the wood stocks can look a little "beaver chewed." As an added historical bonus, many of these rifles have a little name tag beneath the buttplate that lists the name, birth year, assignment, and hometown of the soldier who carried the rifle.