It is because the pressure is coming from all sides equally, and therefore it does not deform an incompressible body. Compare the human body to a balloon full of water. If you apply a pressure of 400 bar to it, its volume decreases with 1.8%, and its height decreases with 0.6%. This is not noticeable with the naked eye. Now put a cube with 1 kg milk on that same balloon, the pressure is only 0.001 bar, but you see the balloon deform a lot. If the human body does not deform, the cells and blood vessels are not damaged.
In engineering this is called hydro-static pressure. Unless the object has a hollow space, it only causes hydro-static stress, which does not damage an incompressible material. This is because the atoms stay in the same place, and stay next to the same atom as before the pressure. If the pressure comes only from one side, like when you crush a cube from the top, then when the pressure becomes high enough, the material needs to move out of the way of the pressure, and can only go to the sides. When the material has to move to the sides, atoms become separated and the material is damaged.
In engineering for metals, the Von Mises stress criterion is usually used to determine when damage occurs. There is stress in 6 directions, but because the material behaves the same in all directions (isentropic), we can use a single number to determine when it will fail. The formula is:
An S with different numbers behind it is a shear-stress. S12 is the shear-stress in one - two direction. They are all zero in this case. An S with two times the same number behind it is a normal stress. S11 is the normal stress in the 1 direction. In the case of hydrostatic pressure, they are all the same S11=S22=S33, therefore the Von Mises stress is zero. This formula shows that a piece of solid steel will not be damaged by pressure, even if you throw it in to the deepest ocean. If the pressure gets high enough, the block of steel will collapse in to a black hole though, Von Mises does not take that in to account.