From looking at the info posted, this thing appears to have two modes: jet (air-breathing) and rocket (non-air-breathing). It smells like they came up with the name first, then figured out an acronym to fit. :-)
But someday, using this new engine technology, it might be possible.
But without an oxidizer combustion is not possible. I don't know of any airplane (jet or propeller) that uses anything but oxygen as it's oxidizer.
Near as I can tell, the article is mixing up two different things. SABRE is being tested as a rocket engine - it recently got some press on here because they had a successful heat exchanger test which, while pretty cool, does not an engine make. As I understand it since that test the ESA has given the nod to SABRE and believes there isn't any technical reason why the engine won't work.
The article mentions that the main advantage will be that "aircrafts can carry less load in terms of on-board liquid oxygen". This is presumably actually referring to rockets, not aircraft, since conventional jet aircraft already use regular old air. "massive throw-away first stages", again, hopefully refers to rockets, otherwise there might be some expensive property damage when your passenger jet drops its first stage on La Guardia.
There is an actual passenger jet piece, though. A separate initiative called the A2, a hypersonic passenger jet based on a derivative of SABRE. If built, it could apparently travel at Mach 5 and take you halfway around the world in something like 4.6 hours, which is pretty close to 4 if you squint.
All of this stuff is being planned by the same company, and Wikipedia says the plane's coming within 25 years "if there is market demand" - so although this all sounds good, and it's nice they've had a successful test, I think it's safe for now to treat this new oxygen-powered jet as vapourware.
(sorry)
I have no aerospace background but this reminds me of ramjets, which have been flying since the 40s but aren't used to transport people.
There's also a documentary on the developers: http://www.youtube.com/watch?v=vZ_a21fPkYM
Or does oxygen-powered mean it requires no fuel?
I'm a bit confused by the brevity of the article...