Is it possible to be a mathematician without being mathematically talented? Apparently I'm not mathematically talented enough to be hired as a postdoc. Is there any way to earn a living while continuing to do mathematics I love (commutative algebra)? I don't like teaching or working in financial market.

Of course you can be a mathematician.  Robert Ammann, Marjorie Rice, and Pál Turán all did spectacular mathematics while a postal clerk, housewife, and Nazi prisoner of war.
Pál Turán was being worked to death under the harshest of conditions in Nazi germany. He worked out results in extremal graph theory in his mind while enduring forced labor. Whenever he could find a scrap of paper, he wrote down some of his results. 
Being a mathematician is a state of mind.
