There is no universal rule here, but there is a reasonable way to think about it that does not depend on who is giving the advice.
The cost comparison
The usual starting point is to compare the repair against replacement. If a repair costs more than about half of what a comparable new appliance would, replacement is worth serious thought. Below a third, repair is usually the better value.
That ratio is a starting point, not an answer, because it ignores how much life the appliance has left.
Age and remaining life
Most major appliances are built for somewhere between ten and fifteen years of normal use. A four-year-old machine with a failed pump has most of its life ahead of it, and repairing it is straightforward value. The same repair on a fourteen-year-old machine buys you an unknown amount of time, because whatever fails next is also nearing the end of its service life.
The question is not only "is this repair worth it" but "how many more of these am I likely to pay for".
What has already failed
An appliance on its third unrelated repair is telling you something. One failure is bad luck. A pattern of them usually means general wear rather than a single component, and the next fault is unlikely to be the last.
What kind of fault it is
Some faults are worth repairing at almost any age because they are isolated: a door seal, a pump, a heating element, a fan motor. Others are effectively the appliance itself — a sealed refrigeration system, a cracked drum spider, a failed main control board on a machine where the board costs most of a new unit.
The things people forget
A new appliance has delivery, installation, and disposal of the old one attached to it, and often needs the space measuring properly. A repair usually does not. On the other side, a new appliance typically uses less water and power, which matters more on something that runs daily.
What we do
The technician tells you what failed, what it costs to fix, and how that compares given the age and condition — including when replacing is the better call. There is no benefit in selling someone a repair that will not last.
Getting it looked at
This is general guidance, not a diagnosis. The same symptom can have a cheap cause or an expensive one, and the only way to know which is to have someone measure it. A technician will find the actual fault and give you a fixed price before any work starts, so you can decide with a real number rather than a guess.