How do 营业, 经商, 贸易, and 业务 fit into the context of other 业-related words?
營業 means your business unit do provide service to others. There is time you need to stop provide service. For example, you pause your service for a while strategically, your employee need holiday, you need time to prepare things before start the service. On the time you provide your service, I can say 營業中. 這家餐廳有在營業(This restaurant provide service), 這家餐廳打烊了(This restaurant have closed). The examples provided above show this different meaning.  So you can know 營業 is totally different from 打烊.

經商 means do trading. If I use tradition Chinese to explain it, I will say 從事商業.

貿易 means trading across different country.

業務 means things you need to do in work. Don't confused with 業務員. It means someone who try to sale things to other.