GA single pilot reported the aircraft's landing gear would not extend; requiring a gear up landing on a nearby grass strip.

Narrative

I had planned and took off from ZZZ for a nice Sunday morning flight on a clear VFR day to get a landing in in the pattern and then fly to another airport to fill up with fuel- I had more than an hour's fuel on board; and was planning to fly to ZZZ1 or ZZZ2- both are very close by; less than 15-20 minutes. I pre-flighted the airplane and ran through my checklists; and nothing abnormal was detected. Upon takeoff; I retracted the gear as normal and flew the pattern as normal; nothing appeared unusual. The plane was flying very well. Upon staring my landing checklists; thought; I was surprised to find that the landing gear did not come down as normal; noting else was unusual; except that they didn't move. At all. I tried the gear again; same thing; except I noticed that my gear handle was now very easy to move- it had not been difficult; but it was now suddenly loose. I got out the manual; and did the emergency gear extension procedure; including pumping the hand pump for the hydraulic system; and putting the emergency gear handle down; nothing happened; although I did determine that I had hydraulic power. Further efforts to lower the gear via the emergency gear procedure; putting it up and down; and anything else I could think of did not move it. Above all; I focused on flying the airplane safely; and trying to find a solution. After quite a while trying to get the gear to come down flying around ZZZ; I flew to ZZZ3 and continued trying to solve the problem. I tried anything I could find in the manual; and contacted a flight instructor by radio and another pilot; and the people on the ground at the FBO for help; and made a number of passes through the pattern to have them see if my gear were actually down; but they were not. After quite a while; I chose to land the plane in a controlled fashion with the gear up; as I could not get the gear to come down. The FBO instructed me to use the grass between the runway and taxiway; and even brought out a car to show me which section; which has been used many times for grass takeoffs and landings before; being smooth and quite long; and with clear approaches and departures; being next to the Runway. I carefully landed the airplane with the gear up on its belly on the grassy area without incident or injury; but with some expected damage to the airplane. I do not know the reason for the gear not working; but it seemed like I never lost hydraulic power; so I do not believe that is it. I believe the unlocks the hold the gear in position when up somehow were stuck; or perhaps something broke that releases them; I was asked about a cable system on Navajos having a similar issue; and I am wondering about that; as none of the three gear came down or released; repeatedly attempting to cycle the gear or the emergency system did not seem to affect it; as if they were stuck or specifically the system to release them malfunctioned. I could have had more fuel that would have enabled me to stay up longer; and continue working the problem; but I am not sure that would have avoided me having to land in a similar fashion. I also could have landed on the runway; but I was concerned that landing there in this circumstance could damage the airplane much more; cause injury; or even spark a fire; following the FBO's instructions matched with my belief that landing on a cleared grass strip or area would be far safer; especially one where the approaches to it were clear and safe; like the area at ZZZ1.