We construct the tensionless limit of bosonic string theory in terms of a family of worldsheets with increasing acceleration and show that the null string emerges in the limit of infinite acceleration when the Rindler horizon is hit. We discover a novel phenomenon we call null string complementarity, which gives two distinct observer-dependent pictures of the emergence of open string physics from closed strings in the tensionless limit. The closed string vacuum as observed by the inertial worldsheet turns into a D instanton in the tensionless limit, while in the complementary picture from the accelerated worldsheet, one sees the emergence of a D-25 brane. We finally discuss approaching the Rindler horizon through time evolution at constant acceleration and also show how an open string picture arises very naturally.