I have to say that this is probably one of the achievements I worked the hardest on, but finally made it. I am not going to go over what the achievement is, since it's about 4 seasons, then Cooperstown, then the legends.
I will start in season 4.
First off, the game will be much quicker, if once you achieve your objectives, you simply bunt all batters. This will eliminate the abilty for the other team to score turbo on strikes, and legendary/great catches.
My hardest challenge here was the one with having to make a legendary catch. Now, there is a trick to this. Since you only have a 5 star rating, you cannot make a legendary catch, unless someone is on the field who adds +1 to your glove. Once you do this, just be patient. My player was at 3rd, and kept throwing curves on the inside part of the plate to righties, and the outside to lefties. It eventually paid off.
Cooperstown is a collection of games, but only a few are tough. Here's some overall tips. First, live with the fact that Ernie Banks can hit. Second, avoid the wheelhouse at all costs. This makes Martinez the one pitcher you don't want to use since it's tempting about how he can shrink the wheelhouse quickly. Also, use your relievers. I pretty much use Scott in the 4th and Tekulve in the 5th. Don't forget, he has a 25K powerup.
The most irritating game is the one where you have to hold a team of sluggers scoreless. This one isn't hard, if you trust your pitching. First, put your player in the leadoff, and Martinez behind him, since there's a +1 for his bat. Lead off the game with a home run, then bunt the rest. Remember, you only need 1 run, since they aren't allowed to score. Bunting, of course, limits their power up points. Again, it's not worth it, so don't go into their wheelhouses.
Also in Cooperstown are the dreaded legendary catch games. I avoided them both. You don't have to win every game, just get enough votes. I chose the shutout game over both legendary catch games.
Once Cooperstown is finished, you will be in the hall of fame. You will then have to do a few challenges. They were basically easy, except I found that the speed one was a pain. Don't worry if you mess up on the Sandberg challenge, he does too. The real trick is not screwing up on the balance beams, and mashing the A button going into the fan/wind. Boggs was relatively challenging, until I focused more on green and yellow balls.
About "The Babe" challenges: I used Ryan for every game, since he can't lose his fastball and it's legendary in more ways than one. Again, steer clear of the wheelhouse and stay with Nolan Ryan's strengths.
Finish the challenges before moving on to the Babe - 4 games, all different. I found the first 3 to be easy, the 4th went extra innings, but won the first time.
The first one, just win. The second one, everyone has a legenday bat.but this was easy if you just keep your cool and stay out of the wheelhouse. The third one is the same as the first, except the Babe has a big slam every time he comes to bat. Really, worst case is you give up a homer twice. Score enough, and it's a non-issue. I saved up a Big Heat and struck out the Babe on 3 fastballs. Incidentally, striking out the Babe in any game is an achievement.
The last game is you against the Babe, but he's pitching. All his pitches are legendary, and you won't be hitting homers (I recommend not using the big slam, save it for heat). The field is massive, and hit an inside the park homerun with my player. I scored another run on a leadoff double, bunt, and sacrifice. One thing to note - the Babe gets a 100K power up in the last inning, meaning the 5th. So, if you're behind, it's lights out. I was able to make it to the 6th, and somehow scored 5 runs on a few inside the park home runs.
Win the 4th game, the achievement pops. You then will go to retro NY to take on the Babe in home run pinball, no achievement, just a litle fun and then it's unlocked. Once you beat the Babe, the credits popped.
Hope you found this one useful.