I think there is no such bijection. A line in the plane is almost the same as a plane through the origin in 3-space (intersecting with the plane at height 1), except there's one plane through the origin that doesn't give you a line (the z=0 plane). So the space of lines in the plane is homeomorphic to $\mathbb{RP}^2$ minus a point: an open mobius strip! So the question is asking if there is a continuous bijection $f:D\to M$ from the open disk $D$ to the open mobius strip $M$.

Suppose there were such a bijection $f$, then $f$ would induce a continuous bijection of one-point compactifications, $\bar f:\bar D\to \bar M$, which would have to be a homeomorphism (a continuous bijection from a compact space to a hausdorff space is a homeomorphism). But $\bar D=S^2$ and $\bar M = \mathbb{RP}^2$ are not homeomorphic.