Yes, it's possible. WillK wrote the basic concept: line up separate units, connect them together and put a runway over the whole. However, this poses huge problems if you use normal ships, because of the way they heave in the waves. Even in calm seas this is a considerable amount of motion (it's hardly perceptible for a human because it's so slow, but that's no help for a plane trying to roll from one ship to the next).
In principle it would be possible to connect the ships rigidly together to suppress this, but that would require infeasibly strong joints, because such big ships have enourmous leverage. Probably the hulls themselves would need to be strengthened too. Flexible joints would be more promising, since they avoid having to fight all the force of the waves – but making those reliable would be a challenge of itself, and it would cause other problems for the aircraft.
Fortunately, the situation if better if the individual units are semi-submersible platforms: those generate most of their lift several metres below the surface, which means they're much less affected by waves.
Another advantage is that these platforms are already designed very heavy, since moving efficiency is much less of a concern than for military vessels. This means they're also less affected by the weight of the airliner. And oil companies already have a lot of experience in connecting such platforms together, so you'll be able to tap into that expertise.
The way I'd design this is like this: first create the runway as a relatively light barge, consisting of beams that are in the lengthwise direction only connected with thin-ish (like, 5 cm) steel plates and a tarmac layer. This way, the structure stays flexible enough so it doesn't need to fight the waves by itself – though of course it also wouldn't be able to support an airliner. Then, after towing it to deep enough waters, you rendezvous with the semi-submersibles, which would either dive underneath the runway to support it directly from below, or attach to the protruding beams on each side. It would possibly be best to make these connections still somewhat compliant, e.g. with air cylinders, but I'm not sure.
After making all the connections, you lift the runway above the level of the waves, either by pumping ballast out of the semi-submersibles, or increasing the piston pressure or with strand jacks. The end result is a runway that is supported strongly from the submerged lifting bodies, but with clearance in between so that the waves can harmlessly run underneath it – at least in calm waters. Whether this structure would be able to survive a storm on the open sea, I'm not sure about.
It would have a lot of similarity with a pontoon bridge. The largest such bridge, Evergreen Point Floating Bridge, has almost exactly the dimensions you envision. It does benefit from crossing only a lake, not open sea – but I don't think that's completely necessary. E.g. Nordhordlandsbrua crosses a fjord: still not open-sea level waves, but they can already get pretty rough in bad weather.
Of course, these bridges are anchored to the sea bed, but sufficiently heavy semi-submersible bodies should get you close enough.