Well, according to this phase chart:
water becomes solid at around 10^10 Pa no matter what the temperature.
So if the pressure towards the center of the hypothetical ocean hit 10^10, one can imagine that a steel bar would stop falling, since it would basically hit something solid.
The next question is how deep a collumn of water would it take to give you 10^10 Pa.
Here’s a chart I found that shows temperature versus pressure of ocean water. (Yes, it’s salt water. Also, at more extreme depths, the line ought to start curving to the right since the water is denser. But this ought to give an upper bound.)
According to the graph, you get 1 dBar of pressure for each meter of depth. So 1000 meters of depth ought to give you at least 100 bars. That’s about 10^7 Pa, according to a pressure converter I found online.
So at no more than 1000km down, the water ought to get solid. That’s about 600 miles.
So my back-of-the-envelope-guess is that the bar sinks about 500 miles and stops.