No, there is no "necessity" for the husband to work outside. The husband can work from home, or maybe due to circumstances be unemployed.
In regards to your question, there are two important points:
1. In Islam, it is wajib for the husband to cover expenses, as nafaqah is the husband's duty. If the wife wishes to pay for things, it is not her shar'i duty, and it is out of her goodness.
2. For the long term, it will not be good or mentally healthy for the husband to be at home all the time, with no work, or future ambitions, or being occupied with at least something useful for himself, his family or for the community. Therefore, it would be advisable that the husband somehow keeps himself busy in this case.
With prayers for your success.