Integrals of products of four modified Bessel functions

We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized hypergeometric and Lauricella $F_C$ functions, and study a number of special cases in which the integrals can be evaluated in terms of simpler special functions or indeed take an elementary form. As a consequence, we deduce some new formulas for definite integrals of products of four Airy functions.