Both QFF and BAEC can earn Sapphire for less than 20K EQP (they are SC based, and there are some anomalies with flight length (10 miles longer flight can triple the SC credit); BAEC in particular has some huge SC earning arbitrage opportunities if you fly F/J or even Y on AA). A few short flights in F make it possible to earn Sapphire for about 5K miles, if I remember correctly (it is some ridiculously low number in miles; substantially more in USD). I'm sure it is possible to concoct a way to do it with less than 1K miles on BAEC (using some of their Bahrain etc. anomalies).

Edit to add an example: 6 LAX-LAS-LAX roundtrips in F earn 720 QFF status credits, which gets Sapphire (for about 3K flown miles); only need 600 SC to retain sapphire in subsequent years. This may or may not be cheaper than 20K ON points on JL. Works on lots of other routes too (LAX-SFO-LAX earns the same SC, but is almost twice as many miles and generally a much higher fare).