Playing Mormon baseball refers to the extremely mild "sexual" acts that Mormons engage in with each other, such as, but not limited to:
Eye contact, light banter, holding hands (NO INTERLACING FINGERS! THAT'S FOR SLUTS AND WHORES), sitting next to each other in church, buying each other promise rings, making rice crispy squares, separated hugging, and, in extreme cases, aggressive spooning.
I could hear Elder Drew and Sharron playing Mormon baseball last night in the other room. I think they were watching 19 Kids and Counting on the couch together!