Quite possibly the greatest television program ever aired by Fox.
The format was pretty simple. Each episode would show six vignettes with supernatural or bizarre occurences. At the end, each story would be briefly recapped and it would be revealed whether each story was fact or fiction. It should be noted that the sole criterion for "factual" stories seems to be a single account, but I'll suspend my incredulity for the sake of a good time.
Most of the stories were kind of interesting and neat like most shows about supernatural phenomena. Original host James Brolin—and later inferior host Jonathon Frakes of "Star Trek"—deadpanned the role and appeared in an all-black stage with various occult-looking artifacts, which gave a nice, mysterious air to it. It was a fine art, but you could eventually learn to guess pretty accurately which were invented for the show. A couple of the fictional ones were laughable—the favorite of me and my friend was a story about a politician who found himself unable to lie in a debate and began his rebuttal with "My opponent is... absolutely correct."
The format of the show also made it a good betting game. My friend and I would put up our stakes then write down our selections on ballots. By the end of this we'd both become pretty good—one of our last games the only difference in our ballots was the first story. I can't recount that without a big dumb grin.
The show was directed by variously by Penelope Buitenhuis, Peter Dashkewytch, Duwayne Dunham, Rachel Feldman, Nicholas Kendall, Caryn Krooth, Robert Lee, Martin Pasetta, Tony Randel (I previously had this as Tony Randall—that was wrong), Skip Schoolnik, Christopher Taylor, and Jeanne Van Cott; and ran on-and-off on Fox from 1997-2002. One website claims it shows on Tuesdays 8:00 ET on the Sci-Fi Channel, but I can't find it in the listings. Take it as you will. Servo5678 says "The show airs on Sci-Fi when they do daily marathons of various shows. In fact, it was just on last week from 11am - 4pm." Again, I can neither confirm nor deny.
If you want to read a summary of the stories and then have the answers spoiled for you, you can check http://www.tvtome.com/tvtome/servlet/EpisodeGuideSummary/showid-3347/Beyond_Belief_Fact_or_Fiction/, because it's way too much stuff to put here and I don't want to ruin the show for anyone.
Sources
Benzite. "Beyond Belief: Fact or Fiction: TV Series." TV Tome. 21 June 2004 <http://www.tvtome.com/tvtome/servlet/ShowMainServlet/showid-3347/Beyond_Belief_Fact_or_Fiction/>.
"Full Cast and Crew for 'Beyond Belief: Fact or Fiction.'" IMDB. 21 June 2004 <http://www.imdb.com/title/tt0138956/fullcredits>.