"various men" is correct and can be used in written English.
You can use it to refer to a group of a variety of men. For example: "Various men gathered in the room to discuss their business plans."
Various men do terrible things to women.
She recalls how mean she has been to various men.
Tells about the duties assigned to various men.
Through the years, various men have joined, one at a time.
"She would introduce me to various men," said Ms. Brawarsky, who had never been married before.
Tells about the various men at the shelter, and writer's impressions of them.
A police report from 2010 says that she was sometimes seen "on rooftops" with various men.
In Rome, various men, including Marius, had hoped for the Eastern command.
Various men have been assigned a heartthrob role: from village beaus to an elderly family friend.
She talked to various men and women who had been gay and then went the other way.
After what feels like an eternity, various men and women run up on stage to claim them.
