I suspect it's a bit of a contradiction in terms. As far as I know, the std RJ45 shells for ethernet expect to see single core wires which by definition are not good for flexing. You may be able to buy braided, very flexible cable but how good it is I have no idea.

I managed to get some surplus drag cable for my machine. It typically has multiple bundles of shielded wires of varying sizes, so you can run sensors, encoders, motors, PSUs etc up one cable without risk of interference, in an oil resistant cover. But as m_c says, the price might interfere with your underwear.

Rapid Electronics "stock" Lapp drag chain cable (https://www.rapidonline.com/drag-chain-cable), although that's "stock" as in they will order it for you from Germany. Possibly worth getting the real deal if you want to make a really pukka job, otherwise use multicore shielded PVC cable inside a conduit etc and use the ethernet / RJ45 for the last few inches in the cabinet where there is no flex required.