In theory, yes it is possible to do so. It all depends on the aircraft type, approach speeds, sea and weather conditions, etc.
During the end of the Vietnam War, a South Vietnamsee refugee landed an OV-1 Bird Dog on the deck of the USS Midway (CV-41). So, yes light airplanes have made unarrested landings on carriers in the past. If your in a light airplane with a stall speed around 30kts and very proficient on your short field landings, this should be no sweat, provided the ship can steam either at, or into the prevailing wind at 30 or so kts. If you're flying a dedicated bush airplane eg an Aviat Husky, Carbon Cub, etc designed for STOL operations, it may even be possible to stick a 3-wire (touch down and hold the main wheels in the third cross deck pendant full stop) under these conditions.
Larger, heavier aircraft have faster approach speeds and need considerably more runway to stop on. The CE-350 Corvalis mentioned above is going to require around 1500-2000 ft to land depending on ambient atmospheric conditions at sea level and calm winds, provided you can fly a short field right by the book. Even with a 30kt headwind, you're looking at an 1100 ft or so of ground roll, making an unarrested landing on a carrier impractical.
If there was no other option available to you, the deck hands could rig the barricade - a device which looks something like a tennis court net on steroids. Granted this device was designed to stop aircraft 10-20 times the weight of a Corvalis at nearly twice the approach speed and I don't know if the arresting gear attached to it have settings available for smaller lighter aircraft. If it doesn't, it could cause serious structural damage to your airplane, possibly making things worse. Maybe that's better than ditching alongside the ship, maybe not.
Another factor would be the sea itself. Carriers are not fixed airfields and in rough seas, the deck lists, pitches, and rolls considerably - think of a 100,000 ton cork bobbing in the ocean - making approaches quite dangerous for neophyte pilots. Rough swellscouldbreak the gear off or do other structural damage during a round out, or cause the pilot to strike the round out, etc. but if there was no other alternative, it's a risk you would have to take.