The engineering of the seals that can operate at those speeds and pressures is a whole specialised field. If I remember correctly, the SSME design uses a labyrinth seal pressurised with Helium.
In other words, the various gases are kept separate only by more gas.
Similarly, the Saturn V main engines were unlubricated because no lubricant could be found that could tolerate the extreme conditions. They were just designed to wear out slow enough to keep operating over their operating lifetime, which was measured in hundreds of seconds (including static test firing).
At those conditions, "lubricant" is probably the wrong way to think about it.
Those pressures, rates of fluid flow, and shaft rpms is going to result in the mechanical surfaces being separated by hydrodynamic forces and/or boundary layer flow.
Technically, lubrication is the same physical principles, but viscosity, weight, chemical stability/ durability and other factors become more important for lubricants that are recirculating.
(Oil in eg, a 4-stroke engine operates on these principles, with the rotating shaft causing pressure differences in the oil that "lift" the shaft from contact with the surrounding metal, until pressure (and therefore spacing) is roughly equal on all sides.
A ping-pong ball floating on a column of air self-stabilizes in the same way.)
The primary concern for these turbopumps becomes heat. So heat transfer and temperature of the "lubricating" fluid become more important than its other nominative qualities as a lubricant. Fluids being pumped at cryogenic temperatures can obviously help here.
So instead of a normal "lubricant", most of these turbopump designs just run a portion of the fuel/oxidizer fluid through the critical areas to provide the surface separation and cooling required.
The documents in the nasa links below don't seem to agree witht this. Actually they are not fluid bearings, they're ball bearings and mechanical surface wear was a big problem for the oxidizer pump bearings in particular because of the lack of good oxidizer compatible lubricants.
Ball bearings require "lubrication", and the lack of good lubricants is exactly why they use fluids that end up behaving as I described, and why "lubricant" is the wrong way to think of it in this case.
If anything, the existence of a good lubricant would prevent the surface wear. Temp control and fluid dynamics are what they have to work with, and yes it results in surface wear because it's suboptimal.
The bearings of something running that fast, under that much stress, would 100% need to be lubricated. Probably the seals are what would have no lubrication and gradually wear out. Look up "labyrinth seal".
If I remember correctly, on the fuel side they use the fuel itself (kerosene) as a lubricant where possible. On the other side I believe they ran dry, but I'm not an expert in the topic.
"Liquid oxygen is [the high pressure oxidizer turbopump's] only lubricant and a poor one at that."
So it's some lubricant and I imagine also significantly a coolant.
This https://ntrs.nasa.gov/citations/20100023061 goes into a lot more detail. But exotic materials and designs bathed in LOX which provides cooling (and minimal lubrication) seems to be how the space shuttle engines achieved their long bearing lifetimes.
The engineering of the seals that can operate at those speeds and pressures is a whole specialised field. If I remember correctly, the SSME design uses a labyrinth seal pressurised with Helium.
In other words, the various gases are kept separate only by more gas.
Similarly, the Saturn V main engines were unlubricated because no lubricant could be found that could tolerate the extreme conditions. They were just designed to wear out slow enough to keep operating over their operating lifetime, which was measured in hundreds of seconds (including static test firing).