While there is an argument in general relativity for acceleration due to gravity not existing we can still explain this using classical physics. When you feel acceleration it is because some of your cells gets pushed into other cells. This is essentially what you feel, the difference in pressure as force is applied to your body. However gravity accelerates your entire body at the same time. You may have experienced that in freefall things fall at the same rate. Essentially all your cells fall at the same rate as well. So you do not feel the acceleration because there is no difference in pressure being applied.
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