The R'Kaal were a fictitious Delta Quadrant species invented by The Doctor in an effort to disguise his attempts to steal the USS Voyager's warp core in 2378. He was being forced to do so by Zet and Nar, two renegade members of the Hierarchy who had captured Captain Kathryn Janeway.
Supposedly, the R'Kaal were ecological extremists who had outlawed warp travel in their space, possessing technology decades ahead of the Federation, including transphasic warp drive and multi-spectral cloaking systems. The territory of the "R'Kaal Imperium" supposedly covered thousands of parsecs, all the way to the edge of the Beta Quadrant.
The Doctor had disguised himself as Janeway and claimed that the R'Kaal had confronted the Delta Flyer II and sent an ultimatum demanding that Voyager forfeit its warp core or face destruction. He also claimed that Voyager had been unknowingly traveling inside R'Kaal territory for three weeks, and that a cloaked fleet was monitoring their compliance.
The Doctor later contacted Voyager as Supreme Archon Loth of the R'Kaal, to ask why they had not ejected their warp core. This transmission allowed Harry Kim and Seven of Nine to uncover The Doctor's fabrication. (VOY: "Renaissance Man")