1.
feeling deeply moved, touched
Describes the warm, emotional sensation of being deeply moved or touched by something. Often accompanied by tears welling up.
2.
tingling, numbness, throbbing
Describes a physical sensation of tingling, numbness, or a dull aching, as from cold, a hit, or a wound.
Two senses
The emotional sense (being moved) and the physical sense (tingling/numbness) both describe a deep, spreading sensation. Context makes the intended meaning clear — 胸がジーンとする is emotional, while 手がジーンとする is physical.
Common patterns
Used with と and する: ジーンとする (to feel moved / to tingle). Also used with くる: ジーンとくる (to be struck emotionally).