The rumors are that they have over 35,000 three dimensional artifacts in storage, as well as, a couple of million items within their library. Unfortunately, due to the lack of space and funding for a new building they can never show everything.Above is a pic of the basement. You can readily see a Joe Cronin autographed ball and various trophies.
It would be a wonderful thing if the Hall of Fame could find a way to display most of what they have. Why not partner up with some Major League teams to sponsor Baseball museums at their ballparks. In fact, since Dodger Stadium is planning their own museum in their upcoming makeover maybe they can strike a deal with the Hall to bring over some items. It could potentially be a mutually beneficial relationship. Consider the cross promotional benefits. The Dodgers would be able to show their fans some great rare items and the Hall could promote their facility and sell souvenirs. It's a win-win.
Hat Tip: VBCF: