While using a lathe can be safer than working with some other power tools and equipment, tools used in woodturning do have dangers that should be accounted for.
Preparing the wood usually requires a chainsaw, bandsaw, or some other cutting tool. One false move can cause the damage or loss of a body part or worse. I permanently damaged my finger while cutting a bowl blank on a bandsaw. I pushed too hard and the blank flew into the blade, along with my hand. I was wearing safety glasses and gloves, by the way, but that doesn’t help improper technique. Learn how to safely and properly use the tools you are working with. Don’t take chances.
Grinders and other tool sharpening equipment can emit sparks and shards of metal. Eye protection is a necessity. Some will recommend against gloves, as they could potential be caught in a grinder.
A lathe can be relatively safe, but there are still dangers such as flying wood chips, flying wood stock, fingers or clothes caught between the spinning wood and tool rest, and the terrifying prospect of having long hair caught in the spinning wood.
Safety glasses are a minimum necessity, while a face shield is more preferable for most woodturning projects.
A work apron can help protect your clothes from flying wood chips.
Dust inhalation is another danger that can be avoided with a good dust mask, respirator, or an dust collector or ventilation system.