A minimax two-facility location problem in multidimensional space with Chebyshev metric is examined subject to box constraints on the feasible location area. In the problem, there are two groups of points with known coordinates, and one needs to find coordinates for optimal location of two new points under the given constraints. The location of the new points is considered optimal if it minimizes the maximum of the following values: the distance between the first new point and the farthest point in the first group, the distance between the second new point and the farthest point in the second group, and the distance between the first and second new points. The location problem is formulated as a multidimensional optimization problem in terms of tropical mathematics that studies the theory and applications of algebraic systems with idempotent operations. A direct analytical solution to the problem is derived based on the use of methods and results of tropical optimization. A result is obtained which describes the set of optimal location of the new points in a parametric form ready for formal analysis of solutions and straightforward calculation.