Even cells that don’t multiply still accept nutrients. Pretty much no atoms in our bodies stick around for too long. It’s why carbon dating largely measures the time an organism died.
In non-replicating cells DNA would be pretty good too. Some of it gets replaced as repairs occur, but most of it would not. I'm not sure that carbon dating would be very accurate over the lifetime of a person (though specific events could cause specific sorts of deposits in bones during the occurrence of those events). And even if it was, radioactive carbon, either because of damage, or because of electrochemical effects, would probably be replaced in DNA more frequently than non-radioactive carbon, as long as the organism was alive, even in non-replicating cells.