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The gnome is near the beginning of the game in the courtyard area that Russell tosses his gun down to you. Look for a white sofa on the other side of a chest-high fence, on the wall with Russ's window. It is partially obscured by a couch cushion but you can pull him toward you with the gravity gloves. You cannot put the gnome away in your wrist pockets however it cannot be broken either.
You will have to carry him with you up until the point that Alyx ascends into the Vault's beam, after the fight with the strider.
This is not a challenge so much as it is a pain-in-the-ass... Save often, especially in areas where there are bottomless pits.
DO NOT leave the gnome in the area outside The Northern Star, even though you return to that area later, the game has several loading screens between when you enter the hotel and when you go back outside and the gnome will be gone.