Considering your average USN ship, say a Tico or a Burke, the size constraint isn't anything all that significant. You only need to get it to fit in a volume roughly equivalent to a good-sized house, for a laser-based air-defense ship. Of course, range, blooming, and other such is a major consideration, but if you're putting enough energy through the thing that you can kill the target before the destabilized air can distort your beam then you're good to go. High-energy low-discharge-time pulse system would work the best, and considering most missiles and aircraft tend to be remarkably thin-skinned well...cutting through three inches of steel in a 1/800 of a second pulse should do the job nicely.
One wonders what this might do for ship construction, though. Cut prop shafts in one pass? Or run enough of a beam splitter that you can laser-cut a bulkhead or hull plate almost instantly...think of something like a giant CNC cutter, and scale way the hell up.