We construct a theory of Maxwell stresses in Rindler space, presenting both a noncovariant and a covariant formulation. The theory shows how the Maxwell stresses are modified by a nonvanishing acceleration of gravity and that the Maxwell stresses of an electromagnetic field produce a volume stress force which vanishes in an inertial frame. In the noncovariant formulation we deduce the Maxwellian force due to the bending of the field lines by the acceleration of gravity. In the covariant formulation we show that Archimedes' law is valid for stationary electromagnetic fields in Rindler space: The Maxwellian surface forces acting from the outside field upon a closed surface produce a buoyancy equal to the weight of the electromagnetic field enclosed by the surface. Generally the mechanism is different from that in a fluid in which the buoyancy is due to a pressure which increases with depth. In a vertical electrical field the buoyancy is due to a tension that increases with height, but in a horizontal field it is due to a Maxwellian pressure which increases with the depth.