In the old days pre-internet, one met members of the opposite sex mostly in three venues: Work, Social events or the bar.

I met my now wife once outside a bar. She was dating a friend of mine briefly. Four years later, when I moved back to my hometown, I took a part time job in a retail establishment (one of three jobs at the time) and she was working there. We hit it off but yes it did get bumpy at one point. I wanted an exclusive relationship and she still seemed to want to date around. So I told her if she wanted to date multiple, fine, but I wasn't going to be one of them. She got mad and accused me of giving an ultimatum. So it was frosty at work for a couple of weeks. But we got back on track and have been married for 36 years.

I think it is possible even today to express an interest in a co-worker if done politely and tactfully. If the response is negative, drop it immiatly. Also, no kind of relationship should be ventured if there is a power-dynamic involved, i.e. one person is a supervisor over the other for example