We investigate the online exploration problem of a short-sighted mobile robot moving in an unknown cellular room without obstacles. The robot has a very limited sensor; it can determine only which of the four cells adjacent to its current position are free and which are blocked, i.e., unaccessible for the robot. Therefore, the robot must enter a cell in order to explore it. The robot has to visit each cell and to return to the start. Our interest is in a short exploration tour, i.e., in keeping the number of multiple cell visits small. For abitrary environments without holes we provide a strategy producing tours of length $S \leq C + \frac{1}{2} E -- 3$, where C denotes the number of cells – the area – , and E denotes the number of boundary edges – the perimeter – of the given environment. Further, we show that our strategy is competitive with a factor of $\frac43$, and give a lower bound of $\frac76$ for our problem. This leaves a gap of only $\frac16$ between the lower and the upper bound.