Bow saw. That is what the curved one is called. And a very light coat of oil---3in1/etc. is fine. Putting protectors on the blade is ok if the blade is oiled and not allowed to dry out---the oil will dry out over time. Rust happens then. Crosscut saws---the usual saw most folks think of are the same. Saws for fine woodworking---dovetail, spinebacked, backcut, and pull saws should not be oiled or have covers. The oil causes all kinds of finishing problems. Just wiped dry and stored so the teeth do not get damaged.

Depends on where you're keeping them, too. If they're in an unheated garage with wide temperature swings, condensation and rust are more likely than if the area was at all climate-controlled.

"Bow saw. That is what the curved one is called. "

Pruning saw.

A bow saw is a metal framed saw with a straight blade held under tension by the frame.

Longest life in a temperature & humidity controlled location.
Pruning and bow saws are used in wet wood and need more care after use.
Woodworking type saws are used in dry wood and require less protection.
For very long term storage oil is used, but for woodworking tools requires removal before use.

Ok, I misread curved.

Many Thanks!!!
Suzi

If you can find a source, camphor oil or camphor oil treated paper stored with the saw will protect it if it's cleaned first (the saw that is). There are sources of the treated paper that manufacturers use to pack with vulnerable items such as steel cutting tools and steel measuring tools but I suspect the quantities you would have to order would be beyond your needs. I found camphor oil at a health food store in small quantities.

Camphor oil, camilia oil, and vapor phase inhibitors can all prevent rust on stored tools, if they are stored in a reasonably sealed container.
Larger tools need to be wrapped in the vapor phase treated paper and then wrapped again in plastic or stored in a closed container (even a zip lock type bag).