Now for my favorite sport: basketball. IMO you could go quite a ways back. The cutoff would be somewhere in the mid '60’s to early '70s for average players.
One cool thing about basketball is that the game is based more on skill and intelligence than athleticism. So guys can play for a long time, against a huge variety of opponents, and show true transcendence. An older player may not be very athletic anymore but he can still show a young up and comer that he has a lot more to learn.
For example, Jordan was able to school Dr. J, Bird, Magic, Drexler AND Kobe, McGrady, and Pierce. Now obviously Jordan wasn’t half the player he was at 38 compared to 24, but it’s still good enough for Pierce or even KG.
After the NBA/ABA merger you can compare any year to any other. I think most NBA fans recognize this. The '80s Boston/LA teams would be the best teams in the league right now, if for no other reason than rules which prevent super teams like that from being formed, but still, the talent is just obvious when you watch the games themselves (as all good fans should – they’re epic, entertaining, and absolutely star studded). Bird/Magic would fight LeBron for MVPs in today’s league.
Someone mentioned Dr. J. He’s not that old, in terms of eras. His prime was in the late '70s but he put up great numbers in the NBA into the mid '80s. He was the 2nd best player on a championship team in '83 at age 32. He’s one of the greatest SFs of all time after Bird and, it looks like eventually, LeBron.
Another thing I wanted to say is that height is pretty overrated by casual fans. For example, Barkley, Rodman, and Ben Wallace are the most dominant rebounders of the modern era and they would be considered woefully undersized for that task. Particularly Barkley, who was just a freak when it came to strength, leaping ability, and desire/eye for the ball. Rodman could probably write a scientific treaty on how to predict where the ball is going to go. Big Ben and Rodman could guard Shaq about as well as anyone can hope to and they give up 4-5 inches and 30-40 pounds.
If anything, wingspan and shoulder height are more important. You don’t rebound with your neck (see Chris Bosh vs. Dwight).
Taking a step back, there are two levels to discuss here in terms of all time comparisons: big men and small guys (i.e. perimeter players, PG/SG/SF).
Big men, across all eras, are relatively equal. Duncan and Shaq didn’t do anything that impresses Wilt. As of right now, 2009, prime Wilt would easily be the best player in the league and it wouldn’t even be close, especially given the paucity of quality big men right now. Duncan/Shaq are well past their prime (Shaq is 38) and KG is getting there + he’s injured. Dwight is unrefined offensively and has huge troubles against thick guys like Yao and Perkins. I’d pay to see Dirk or Amare try to cover Wilt. After that there is a huge cliff.
You can see this smooth transition in what may be called the daisy chain of big men. In no instance did a new generation come and cream their aging predecessors. Mikan to Wilt and Russell, Wilt to Kareem, Kareem to Hakeem/Moses, Hakeem to Shaq, Shaq to Duncan/KG, then to Dwight.
Kareem, in particular, played against a huge variety of players because he played for such a long time – '69 to '89. He was able to block the shots of both West and Jordan, which is kinda crazy to think about.
If anything, the amount of quality big men peaked in the late '80s/early '90s. Look around the league today and you don’t see many quality big men. This is partly due to style/rule changes, but mostly due to a lack of talent. Big Z, Vlade Divac (in 2001, at age 32) and Magloire can make the all star team in our era. Think about that. Duncan and Shaq don’t care about rule changes, they still win titles, and any of Hakeem/D-Rob/Ewing/'Zo (before injury) would be feasting on today’s softies.
For the perimeter players though…they have evolved a tremendous amount. I don’t think it’s hyperbole to say that 70% of the perimeter players during the '60s and early '70s couldn’t make a roster today. Many couldn’t dribble left, they dribbled with their head down, they sucked at defense, weren’t that athletic (played mostly below the rim), couldn’t shoot at range, had low percentages in generals, and rules prevented them from practicing flashier stuff that has been popularized over the last 25 years. Of course, most players wouldn’t be able to dribble for more than 5 seconds in the '60s before being called for traveling or palming, so perhaps the old players could learn new tricks. But current players have been doing this stuff since they first picked up a ball.
Now, you could make arguments like well percentages were lower back then because it was a track meet style of play and the pace was insane compared to today (which is true). Teams just took crazy shots before the defense could set in, which is a good idea because penetration was heavily discouraged for a variety of reasons. But all you have to do is watch tape of the finals from the '60s. You’ll laugh at the primitive perimeter play.
But still, some guys were just spectacular. Jerry West (The Logo) would be one of the better guards today, probably behind LeBron/Wade/Kobe but he’d be right there. Oscar Robertson was like a '60s chubby version of MJ and indeed had the best guard post play until Jordan came around. Tiny Archibald was a bolt of lightning who has the distinction of being the only player to lead the league in both ppg and apg in the same season.