Without currents, I hope these calculations are correct...
Assume 1 mm^2 cross-sectional area stainless steel wire. 4000m (4km) of wire has a volume of 4000 x 1000 x 1 mm^3, or 4000cc^3.
The density of SS is 8gm/cc^3, so the wire weighs 4000 x 8g, or 32kg. The tensile strength of SS is about 500Mp, which is approximately 50kg/mm^2.
So that 1mm^2 SS wire would just about support 5km of its own weight. In seawater, 4km of wire would weigh less than 32kg, so that reduction would work in its favour.
Once you start factoring in currents, things start working against you. It might require, for example, 2mm^2 wire to allow for currents and any wind on the surface, but it could still be done.