Because they don’t put any pressure on them.
Take the TOW missile. It has two wire spools on the missile itself (sufficiently separated to the side from the rocket motor that the wire doesn’t end up inside the thrust jet) and as the missile flies the spools distribute more and more super-thin wire. The wire only has to be strong enough to drag out more wire from the spool (which is of course designed so that there is minimal resistance).
The reason why there are two spools is to make the drag/weight on the missile symmetrical (the wire does pull slightly and weighs a bit) and two wires mean that if one wire breaks the other can still send the control signal.
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