So, the "right to own things," is probably more commonly called "property rights." There's some legitimacy there, but this notion is far from absolute and should be subject to reasonable regulation. I can't just own a nuclear power plant, despite my distaste for the public electric utility.... There are risks, in this exaggerated example, to other people and their property--and other--rights. Rather, I could own a nuclear power plant, if I could afford it, and I complied with regulations on that ownership/operation.
Nuclear powerplant, gun, car, dog, the common thread is that while I can own them all, I must take steps to avoid imposing on others, no matter what that imposition is. The responsibility scales with the potential imposition on others. I have to have my dog licensed and all the shots up to date. Insurance on a car.... Registration and a working safety on my gun, etc.... See the sliding scale implied with the different types of objects owned? Hey, even the NRA is allegedly is all about "responsible gun ownership." Do I think every idiot out there should have the power of life and death over everyone else with a point and click interface? Not really, no. There've got to be some limits on it. I think we can actually all agree that there should be some limits, unless people favor giving convicted armed robbers guns or equally silly things, but the question is where those limits should be.
Property and ownership rights. Yes. Limits on those. Yes. Otherwise my neighbor might think a surface to air missile would look great next to the hedge.