Kimio - there can be a huge difference between deer and elk and therefore the calibers and bullet weights/ballistics involved. Since you are a senior poster, you must know a fair bit about guns and ballistics.
First, you didn't describe what you consider to be affordable. Is it closer to $500 or $1,000 or $,1500? You can get the job done at all three, with the right caliber, decent optics and close-enough range, but at $500 new it might be a single-shot primitive-hunter-eligible rifle (like .45-70) with a low-power scope, good for close-in shots, or you could buy a very decent used rifle and hopefully-good scope. Next, you didn't quite say that your dad has never hunted anything before, nor what his shooting experience and ability are.
It would be helpful if you describe your dad's previous hunting or shooting experience, to elicit more responses. Also, will he be hunting with experienced guides or seasoned hunting companions? Will he be hunting in icy-cold temps? All of that matters. The only addition I'd make to Creeper's comment is that I would not hunt elk with a .243, period. I'd be surprised if that were allowed in the state. Sure, it can be done, but there is so much blank space here, including that your dad has never hunted before, that a .243 is no-no-no in my book until I know that he has been to Camp Perry and also killed 1,000 prairie dogs. He might be a crack shot on paper, but that is not the same as placing a light-weight bullet in the right place on a big, strong animal. With a cartridge like that, there is a good chance of wounding it and losing it.
Don't forget to scour sites like Gunbroker for good used big-game guns, like No. 1 Rugers, that function well in really cold climates where some of the modern ones might want to freeze up. Given the range of game you mentioned, elk and deer, I would not go out with less than .308 or .30-06 and more likely would tend toward 7mm Magnum or .300 Weatherby or H&H.
All that said, good for you that you are helping to get your dad out in the field. It is an uncommon reversal of roles, but a welcome one nonetheless.