We need a place (that/where) we can stay for a few days.
Might I trouble you to tell me whether 'that' is also possible in the question? I think 'where' alone is correct because 'stay' is an intransitive verb. 'Where' can be paraphrased into 'in which', but 'that' is just 'that'.
We need a place in which we can stay for a few days. (O)
We need a place that we can stay for a few days. (?)