An interesting bit of history: back in 1983 there was another Congressional Page sex scandal, in which one GOP lawmaker confessed to relations with a female page in 1980, and also in which Gerry Studds -- a Democrat from Massachusetts, the first "out" Congressman, and posessor of an apt name for this story -- admitted to a relationship with a 17-year-old male back in 1973.
Studds won re-election and the Republican did not. Most of Studds' constituents already knew he was gay, and he basically didn't apologize for what he did:
Studds, however, stood by the facts of the case and refused to apologize for his behavior, and even turned his back and ignored the censure being read to him. He called a press conference with the former page, in which both stated that the young man was legal and consenting. Studds did not break any U.S. laws for that time, in what he and page called a "private relationship."[1] He continued to be reelected until his retirement in 1996
Bill Clinton could have learned quite a lot from this guy.
Now, let's be clear. Being an adult and getting intimate with a 17-year old is ethically dubious. It's almost certainly unethical if there's a workplace power relationship. If you happen to be considering trying something like this, don't.
However, I think the response of Studds shows the power of standing behind your choices. I may find Studds and the 17-year-old iffy, but if the 17-year-old is willing to stand by his man ten years later and say it was all good, that's a strong statement.
This requires you to embrace your actions in the first place, of course, and to have a solid internal moral gyroscope you can live by. But if you've got those things and a clear head, you can potentially break a lot of social rules to little ill effect.