She used to live next door to me.

Literal

She [topic-は] my house [possessive-の] next [at-に] was-living.

隣 ('next, adjacent') typically refers to immediate proximity — the house next door, the seat next to you, the table beside yours. The compound 私の家の隣 ('my house's next') stacks possessives to specify the location precisely. The past progressive ~ていた here often translates to English's 'used to' for habitual past states — she lived there over a stretch of time, not just for a moment.