The rough guideline most technicians use: if the repair costs more than about half the price of an equivalent new machine, and the appliance is past the middle of its expected life, replacing is usually the better spend. It's a starting point, not a law, and it's worth knowing where it breaks down.
Expected life varies more than people assume. Refrigerators and gas dryers commonly run well past twelve years. Dishwashers and front-load washers tend to have shorter working lives, and window air conditioners shorter still. A ten-year-old refrigerator with a failed start relay is worth repairing; a ten-year-old dishwasher with a failed wash motor and a leaking door gasket usually isn't.
Which part failed matters as much as the age. Control boards, sealed-system compressor work, and drum bearings are the three that most often tip a machine into replace territory, because the parts and labour together approach the cost of a new unit. Pumps, valves, heaters, thermostats, relays, seals, and belts almost always come in well under that line.
There's a quieter factor too: build quality. Some older machines are genuinely better made than what replaces them, with simpler mechanics and parts that are still available. A well-built fifteen-year-old washer with a fifteen-minute repair is worth keeping. A cheap five-year-old one that has already needed two visits is telling you something.
Our own rule is simpler. If we wouldn't spend the money on it in our own kitchen, we say so before we start — and we'll tell you what we'd buy instead. Losing a repair job is a much smaller cost than a customer who spent three hundred dollars to get six more months.