Whether or not Wyatt Earp owned the famous "Buntline Special" Colt Single Action Army Revolver is the subject of great debate, with little evidence proving either case. He may of had one, but there are no records or documentation available.
Wyatt Earp is not depicted wearing a mustache. He was known to have had one since he was a teenager, which went all the way up to his death in 1929.
Wyatt Earp was 5.5 years older than Bat Masterson. The difference between Hugh O'Brian and Mason Dinehart (who plays Bat) is 11 years.
All of the buildings in the series were clearly manufactured using modern carpentry equipment. The lines are too straight and none of the wood is warped or has knots in it.
Most of the male characters (main, recurring, and guest stars) in the series are clean-shaven which was not the fashion at that time.
The rifles that are used throughout the series are Winchester 1892s. The show ends in 1881 making its use anachronistic.
The female characters in the series wear makeup, have hairdos set using hairspray, and they all have manicures. None of those things were available in the 1870s and most were only fashionable following WWII.
The real Wyatt Earp wore at least a mustache from his teens, until his death in his 70s. The actor portraying him in the series was clean-shaven for the entire run of the program.