A binary sequence satisfies a one-dimensional$(d_1, k_1, d_2, k_2)$runlength constraint if every run of zeros has length at least$d_1$and at most$k_1$and every run of ones has length at least$d_2$and at most$k_2$. A two-dimensional binary array is$(d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)$-constrained if it satisfies the one-dimensional$(d_1, k_1, d_2, k_2)$runlength constraint horizontally and the one-dimensional$(d_3, k_3, d_4, k_4)$runlength constraint vertically. For given$d_1, k_1, d_2, k_2, d_3, k_3, d_4, k_4$, the two-dimensional capacity is defined as $$displaylines C(d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4) hfillcr hfill=, lim_m,n rightarrow infty log_2 N(m, n ,vert, d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)over mn $$ where $$N(m, n ,vert, d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)$$ denotes the number of$m times n$binary arrays that are$(d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)$-constrained. Such constrained systems may have applications in digital storage applications. We consider the question for which values of$d_i$and$k_i$is the capacity$C(d_1, k_1, d_2, k_2; d_3, k_3, d_4, k_4)$positive and for which values is the capacity zero. The question is answered for many choices of the$d_i$and the$k_i$.