How can I get rust stains off of a 30 year old linoleum floor without using chemicals??

Without using chemicals, your only choice is lemon juice and cream of tartar. Make a paste and apply it to the rust stain, agitate and allow to dry. Sweep up afterward. It may take several applications. If you do want more instantaneous gratification, CLR, WINK and Mostenbockers rust stain remover will do the trick.