The DNA molecules contain phosphate groups involved in $3'\rightarrow 5'$phosphodiester linkages. These groups, in the bonded state with deoxyribose, contain 1 negative charges ($\ce{3'-PO4^{-}-5'}$). This means that the two strands are very highly negatively charged. What than stabilizes the double helix, preventing the natural electrostatic repulsion between the two strands?
[I am assuming that the phosphate groups are dissociated under physiological pH, correct me if I am wrong.]