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Materials science undergraduate student here (not a mechanical engineer, my knowledge is limited in the area, I did not go to great lengths to ensure I'm right here, etc.).
A typical method to generate high pressures in research are diamond anvils. This is suitable for exploring the behavior of cells and microorganisms under high pressure.
For human preservation, however, you'd need a pressure vessel. As the yield strength of your typical steel is on the order of 100, maybe 300 MPa, you're really up against a wall here, materials-wise. I don't doubt that suitable alloys for human-sized pressure vessels at 350 MPa exist, however, such vessels will be expensive, and controlling processes within will be difficult. In any case, generating such pressures will probably not involve a moving piston.
I can't really tell whether or not the procedure you've outlined is viable, but I'm quite sure it's far from trivial, just from an engineering point of view.
The concerns of user passive_fist are also valid.
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This is definitely an interesting idea but there may be a lot of unforeseen problems. At pressures as high as 350 MPa, chemical reactions that are energetically unfavorable under normal pressure suddenly become favorable (this is why, among other things, air becomes toxic). I have no idea how this would mess with the various chemical processes that go on in the body, but my guess is that unwanted reactions could lead to development of fatal toxins. Also, just as such high pressures cause phase changes in ice, they could cause phase changes in plenty of other molecules.
I'm sure someone who is more knowledgeable on this could weigh in, but googling around a bit revealed some studies on subjecting human cells to high pressure. This study subjected human amnion cells to 70 MPa and found significant changes in cell activity involving blebbing (this is when the cell membrane disassociates from the cytoskeleton), although they did point out that the effects of pressure were reversible, which is promising.
Hmm, I wonder what the exact biochemistry that prevents life forms (including, apparently, vertebrate fish) in Challenger Deep at 111 MPa from experiencing these problems is, and whether it can be replicated in mammals.
They also mentioned that blebbing first appears at 90-120 seconds, but that's way too short even for the fastest protocols possible. Theoretically, it's not unthinkable to cool the body to just above 0C, and then go straight to 632 MPa and above, to make it instantly freeze, before blebbing occurs. And then, if total liquid ventilation allows one to drop the pressure that quickly as well, just go from solid directly to a non-dangerous pressure range. But for any protocol that involves temperature changes under pressure, tens of seconds is positively too short to allow the temperature to stabilize.
As for toxicity though, I though it was entirely due to the increased partial pressure of oxygen (which thus creates too strong of an oxidizing environment) and having too many nitrogen atoms dissolved in tissues, physically messing with fine-grain biochemistry like ion channels. Is there another chemical component of toxicity beyond that?