Per Elon Musk at 22:04 in the SpaceX "Starship Update" Youtube video, he states "the best design decision on this whole thing is 301 stainless steel". Type 301 is an austenitic stainless steel contains up to 0.15% carbon, 16%-18% chromium, 2% manganese, and 6%-8% nickel (the rest is Iron and a few trace elements). 301 has been around for a long time and has been commonly used since at least the 1960s. It is primarily used in the cold worked condition. It can be strengthened by cold work to an exceptional extent.
For the Starship 301 SS, SpaceX is probably keeping a tight control on the Nickel content which greatly determines the ability to cold work the steel. They are then cold working the steel followed by stress relieving. I'm not sure to what degree they are cold working the steel, if you go too far it starts to get brittle. However, even in the Full-Hard state, with an tensile ultimate strength of 180-220 ksi, it still has a failure strain of about 9%, which is not bad. As Elon states, it does have pretty impressive cryogenic properties. At -400F, it has about 230% of its room temp tensile ultimate strength.
More info can be found in:
MMPDS Chapter 2 Steels - AISI 301
Aerospace Structural Metals Handbook Vol 2 - Code 1301