Does that make sense? I don't see how pressure can make things weigh more... but I guess 10km of steel cable will have some weight. I'd have thought you'd make the cable have the same net density as water, so it is effectively weightless.
I don't know the specifics of how the steel/iron/other compounds would react to that pressure but that's a guess.
Edit:
Other sources leave out the pressure part, it looks like the weight of the cable would be the issue. I guess it's also why we don't have a space elevator yet.
It still has momentum. So it could be broken just by trying to pull it - even without the weight.
And you are right - "pressure conditions" is not correct. It's simply the weight of the cable.
There is no void (air) inside the cable, and the pressure is even all the way around - the only crushing you can do is move the atoms closer to each other. (Compress actually, not crush, and iron is basically incompressible.)
Most metal objects do actually have voids in them, but you can manufacture metal objects that don't, and it's not clear that having voids in your cable would cause it to be damaged by pressure.