Use the adjective run-down to describe something that's so old it's falling apart, like a beat up old car or a worn out mattress with springs sticking out of it.
A dingy neighborhood that's seen better days can be called run-down, and so can your decrepit apartment building, with its broken elevator, creaky stairs, and windows that don't close all the way. A person can also be run-down, if he is bent and elderly, or just exhausted from traveling for days or working too many late nights. You can also spell it rundown, although the hyphenated run-down is more common.
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