When you're lonesome, you feel sad about being by yourself, and you miss the company of other people. It's entirely possible to feel lonesome even when you're in a crowd.
Living alone on an island might sound appealing sometimes, but it would probably be pretty lonesome in reality. Good elementary school teachers pay attention to which kids look a little lonesome during recess, especially at the beginning of the school year, and match them up with a pal. Lonesome means "lonely," although it's uncommon outside of the US.