According to the Cambridge Dictionary
be staring at something
If you are staring at a bad situation or problem, you know you must
experience it or deal with it very soon:
By late in the first half the team was staring at almost certain
In the Oxford Dictionary, we found a related phrase
be staring something in the face
Be on the verge of defeat, death, or another unpleasant fate.
In your sentence the unpleasant fate / bad situation is starvation. That family is going to face starvation soon if their problems are not solved.