Atari never went out of business, but it has changed hands a crapload of times. The original Atari company (which included arcade games + home consoles/games) sold off the home business to Tramel Technology in 1984, then Tramel Technology renamed itself to Atari. In 1996, Atari merged with JT Storage, then JT Storage sold Atari off to Hasbro Interactive in 1998, then Infogrames bought Hasbro Interactive in 2001, and several years later, Infogrames renamed itself to Atari. There was a bankruptcy filed by this company (or at least its US division) in 2013 but it emerged from it.
As for the arcade games part of the original Atari company, it was split off as "Atari Games" (not to be confused with plain old "Atari"), then was sold to Namco in 1985. Namco sold it off to a group of Atari Games employees in 1986. Time Warner bought it out in 1993, but then sold it to WMS Industries (owner of Bally/Midway) in 1996, and they renamed it to "Midway Games West". Midway Games West was disbanded in 2003, but its copyrights/trademarks were sold to Warner Bros in 2009. So the arcade game half of the original Atari is actually dead (unless Warner revives it someday), but the home console/games half has never been.
Edit: on another note, "Ataribox" is a really dumb name. We had the Atari 2600, 5200, 7800…the new one should be named the Atari 10400, or maybe the Atari 10.4K.