The hydraulic press generates those forces between two points. Everything above or bellow that is not taking any of the load of the press. Think of it as a machine which squeeze things. The fact that most hydraulic presses are vertical is just not important, they could just as well be mounted horizontally. A hydraulic press which can press 100s of tons needs to be built to handle these forces of course. This is why hydraulic presses tends to be made out of thick solid metal beams. And steel is strong enough to take all that force in tension. But you can use a hydraulic press on top of a milk cart if it can handle the weight of the press itself as no forces go further down then the bed of the press.
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