Hard to say. I'm skipping the "air, water, food, etc..."
I'd be pressed not having a way to to play music. Music is very important to me.
But if I had to choose one object to live with/walk around with, I might choose a camera. I could go on about why, but won't.
If I had to choose one thing I have that does/would help me get by in the world, I would choose my iPhone or a laptop. Reasons seem obvious. These are the most important objects in my day-to-day life.
If I had to choose one sentimental object, I would probably choose a book that my father gave me (a book that was his as a child). Possibly a photograph that he took for me. Or possibly an old press photograph of my grandfather that I really like/matters to me. But I think the book.
I'm thankful that I probably won't ever have to make the "one-object" decision, but I like thinking about it.