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They do nothing at all. If by closeted you mean gay men who are not out, I have known quite a few, and they do nothing aside from simply not stating that they are gay.
Well, the one thing they will often do is have a girlfriend. I have no idea how this relationships works.
Other than that, they do not try to hide their homosexuality. They aggressively pursue men, especially straight men (like me). Yes, I have been pursued by quite a few closet cases, and I have a low opinion of them.
One things they don’t do is feign an interest in women. The closeted gay men I know never talked about women, not even once. They never looked at women, not even one time. In fact, that is one of the ways you can figure out that they are closeted if you watch them closely over a period of time. Look at how the man looks at and talks about women. If he never talks about women, and he acts like they are ghosts, he may well be closeted.
I had a closeted gay boss one time who acted like the beautiful women in the office were part of the furniture. He reacted to them the same way you would react to a chair or a desk.
I knew another closeted gay man at another job. We would eat in the lunchroom, and a hot woman at work would walk in the room, and every man would look up and stare at her. All except for Arthur. To Arthur, those women may as well have been ghosts. It was as if they were not even there.
There was another closeted gay man at the same work. I had no idea he was gay until a co-worker once told he thought the guy was. He said the man got phone calls at work all the time, but they were always from men. Now that I think back on it, I am quite certain that this man was gay.
They were all very easy to figure out, as closeted gay men continue to pursue men sexually, and gay men are quite easy to figure out if you have been around them. For one thing, they give off a strong sexual vibe when you are around them.
One thing I have noticed is that closeted gay men are often not effeminate. The three men described above were all wimpy but not effeminate. They were much wimpier than the average wimpy straight men. In fact, they were so wimpy that it was almost bizarre, a parody of wimpiness. They acted like mice.
There are some other giveaways. I had a boss who I strongly suspected was a closeted gay at a job I had. One day he opened a drawer, and something happened that shocked or surprised him. For some reason, he shrieked in a very high pitched girly voice due to shock. All of the women in the office were looking at each other and me with raised eyebrows. You would almost never see a straight man shriek like that, especially over something trivial. I later figured out that this man was quite closeted.
The only thing different about closeted gay men is that they are not out. They simply do not state that they are gay. I am not sure what they would say if you asked them. One closeted gay men threatened to beat me up when he found out I was telling people he was gay (he had been pestering me forever and would never take no for an answer). Honestly everyone knew he was anyway.
The case of closeted gay men seems to be the “open secret.” It’s something that everyone knows, but no one ever talks about it. I recall once at my aunt’s workplace, a number of closeted gays at work got together one and aggressively came out in front of the whole office in a pretty in your face way. I assume they were looking for a bad reaction. Instead, the whole office simply shrugged and everyone said, “Oh well, we always figured you were gay anyway.” The gays were disappointed, as apparently they were looking for an extreme reaction.
Some of the closeted gay men I have known were profoundly unhealthy psychologically. Even most gay men seemed healthier than these closeted men. Some of them were seriously screwed in the head in a bad way.