In this study we develop a formalism to describe the organization of DNA inside bacteriophage capsids during genome packaging. We have previously shown that DNA inside bacteriophage phi29 (phi29) is organized into folded toroids [A. S. Petrov and S. C. Harvey, Structure 15, 21 (2007)], whereas epsilon15 (epsilon15) reveals the coaxial organization of the genetic material [A. S. Petrov, K. Lim-Hing, and S. C. Harvey, Structure 15, 807 (2007)]. We now show that each system undergoes two consecutive transitions. The first transition corresponds to the formation of global conformations and is analogous to a disorder-order conformational transition. The second transition is characterized by a significant loss of DNA mobility at the local level leading to glasslike dynamic behavior. Packing genetic material inside bacteriophages can be used as a general model to study the behavior of semiflexible chains inside confined spaces, and the proposed formalism developed here can be used to study other systems of linear polymer chains confined to closed spaces.