Seems doable to be honest—huge spools of undersea cables are deployed on ships all the time.

A spool 20 feet wide has a circumference of ~60 feet, and 3 miles of cable is ~15,000 feet, so 250 loops around the spool. Seems doable.

Using 9mm diameter semi static climbing rope as an example.

For volume: Volume of 4km of rope is 0.009^23.144000 = 1.02m^2 which seems like an amount that would fit on a large spool on a ship.

For weight: rope itself weighs about 60g per meter so 0.06*4000=230kg of rope. It has a breaking strength of 22kN, which is roughly 2000kg static load. Given everything is probably roughly neutrally buoyant seems like enough.

Not saying its practical but it seems like the actual volume/weight of rope would not be a problem.

You’re aware of ROVs? They usually communicate with an umbilical, which is considerably fatter than a piece of string. It’s deployed from the surface vessel, not from the ROV. The same could just as easily apply.

Couldn't it be that the spool with the rope floats on the surface and only one end of it is connected to the submarine with a snap hook? It may work and saves me the maths exercise lol :)

String would not work, it's not strong enough. Steel cable would likely work. Depends on tensile strength of the material.

Do the math for the weight of a string 2 miles long.

Then take a look at how much weight a string can hold up.

Then you're ready to be taking on space elevators