The column of liquid in the barometer imposes a downward force (weight) = mg, and the pressure at the base of the column is rho * h * g (or rho * g * h). This pressure is balanced by the atmospheric pressure pushing down on a pool of liquid at the base of the column or end of the liquid column if there is a 'U' bend in the barometer (the sealed end (above column of liquid) of the barometer should have a vacuum).
When the elevator 'accelerates' upward, the column of liquid experiences an effective weight of m(g+a), producing a pressure at the base of the column of rho * (g+a) * h. When the elevator 'accelerates' downward, the weight is reduced, m (g-a).
In free fall, the elevator is accelerating at the rate of g, so the column has no weight and the atmospheric pressure pushes the column up into the barometer and there is some pressure on the container. But it is not infinite - it is limited by the pressure of the atmosphere pushing on the pool of liquid at the base of the barometer.