Can I apply Fourier Transform to a non-time-indexed signal?
Yes you can. The unit of the "frequency" axis after the transform will be $m^{-1}$, and is known as spatial frequency.

For example, if there is a strong peak in the Fourier transform at $3.10^{-4} m^{-1}$, it means that your original curve exhibits a strong pattern that repeats at a scale of every $3.3km$, and from that you could infer that maybe the signal was recorded from a vehicle doing laps at the Monaco Grand Prix. The harmonics of this spatial frequency would contain a "signature" of the shape of the circuit.

A practical application of this is handwriting recognition - looking at shapes in the Fourier domain yields representations invariant to scaling, rotations, or more robust to deformations than the original data.