You can play some of the D&D board games solo. There are solo scenarios in Wrath of Ashardalon and Castles Ravenloft.
Toward the back end of the 4E DMG, there is a section on playing D&D without a DM that you might want to check out. It's essentially about how to put together an appropriate encounter deck and map deck to create random dungeon encounters. It would probably run better with at least two people than just yourself, but it'd be a start.