Step 1: Meaning of the sentence.
The sentence says that the speaker had already planned to go skiing but had to cancel at the last moment due to an injury.
Step 2: Analyze each option.
(A) \emph{back up} → means to reverse or support something. Not correct in this context.
(B) \emph{back of} → not grammatically valid here.
(C) \emph{back on} → means to rely on something. Does not fit here.
(D) \emph{back out} → means to withdraw from an agreement or plan. This perfectly matches the idea of cancelling the skiing plan.
Step 3: Final Choice.
Hence, the correct option is: \(\boxed{\text{back out}}\).