rtdonell1: the Saiga's come in a 16.5 and a 21.5" barrel length. One of the fellas on the Mil-Surp forum has one and raves about its handling and firepower. I don't think he has specifically mentioned the accuracy of this rifle thought so I don't know if this could or would be your 500 yd rifle - probably not. Even the M1As, the civilian version of the M14, aren't all that accurate at 500 yds unless you get a target or sniper version and then you are gonna spend some serious dollars - way about the $300 dollar mark you mentioned.
You also didn't say whay you preferred a semi-auto over a good bolt action. As I recall, you have to spend some serious $ for that kind of accurate semi-auto, but there are many military surplus rifles and calibers available that are absolutely excellent for what you would like to do, and they are available for under $300. One that immediately comes to mind is the 6.5 x55 Swedish Mauser. You can find a plain jane short rifle with a 23" bbl for under $300 and one that may be in excellent condition with a great bore. You can also find target models and snipers but they are a bunch more expensive. With the 6.5 Swedes, some minor stock and trigger work should get you a sub-1" group at 100 yds. Some of the Moison-Nagant rifles in the 7.62x54R Russian round can be had in the original sniper version and they are all well spoken of. Same with the 303 Brits and finding one in good condition. Also, the 8mm Mauser in either the M98 or M48/48A configurations. As for capabilities, either of the rounds mentioned are very capable big game cartridges for what you wish to hunt.
Yes, the 308 and 30-06 are good long range rounds, but you would need to handload to get 500 yd accuracy, especially on the 'All-American' budget (been there and haven't left yet, lol). Of the other calibers mentioned, target or accuracy loads are also available.
I would start with a bolt action first and see how you shoot with it and whether you are comfortable with it or not. Later, when things improve a bit you may wish to look at some really accurate semi-autos depending on how ya'll feel. BTW, if you head on up to the Mil-Surp forum you can read a bunch about the different calibers and their accuracy. If you look at the thread about the Postal matches, you will read about some of the group sizes these old warhorses turn in - that might help you make up your mind. HTH. Mikey.