I have worked on a cartesian robot with similar requirements† as your own, and we selected direct drive synchronous linear motors for our x/y stages. In our case, both axes were around 2m in length, but magnet the tracks have the potential to be as long as you need to build them.
† Less than an order of magnitude higher at $2/5ms^{-1}$ velocity & $10/20ms^{-2}$ acceleration for x & y stages respectively.
In our case, we used an exposed magnet track linear motor from aerotech, but for most applications a u-shaped track is more convenient as it contains the high magnetic fields within a more enclosed space. Our cartesain robot had large exposed magnet track surfaces which meant that we had to post warnings regarding letting pacemakers and metal tools getting too close (a screwdriver could be attracted to the magnet track with sufficient force to damage both and anything which got in between, such as your fingers).
There are a number of manufacturers making direct drive linear motors (as a google search will show), and in addition to flat and U-shaped tracks, manufacturers such as Nippon Pulse and LinMot do cylindrical shaft motors, which might be easier to retrofit to a robot which currently uses a lead-screw linear actuator.
The accuracy of these direct drive systems can be very much dependent on tuning. They often suffer from significant cogging effects, though some manufacturers claim to include patented zero and anti-cogging techniques and often maximum and minimum speeds can be related to how smooth you want the movement to be at those speeds.
As an aside, we also used air-bearings on our X track, which is how we managed high speed and acceleration with a relatively small motor. They were really difficult to tune however and this was complicated by the fact that we had 6 X-carriages on each Y beam (of which there were two, so 14 x/y axes and 16 motors in total).
