Yes, absolutely you are eligible to get a government job in the teaching line in school. You have already completed post graduation and completed B.ED course which is prerequisite for teaching as Post Graduate Teacher (PGT). With this qualification you can apply in government as well as private institutions for teaching as PGT teachers.
As far as TET is concern, you can appear in TET exam. TET exam are conducted by CBSE is called as CTET and state level exam is STET. I will suggest you to appear in CTET instead of STET as CTET will be valid in all states. But CTET exam is conducted by CBSE and paper level is also tough hence side by side you can also appear in State TET. After qualifying CTET exam you will be eligible to apply for primary as well as Junior classes or uptp 10th Standard. Next CTET 2020 will be conducted in June or July month. Keep checking website of CTET for notification. You can apply for vacnacies in Kendriya Vidyalaya, Navodaya Vidyalaya, DSSSB, or other State level teaching exam. You have Master Degree with Biochemistry subject, hence you can apply as PGT Biology teacher and after qualifying CTET you can apply for TGT Science teacher.
But as you are having a Post graduate Degree in Biochemistry, you can also take National Eligibility Exam (NET) conducted by CBSE or State Level Eligibility Exam (SLET) and can apply for job of lecturer in degree colleges and University.
You can also join Ph.D. after qualifying NET or Research Eligibulity Test (RET) conducted by several University for admission in Ph.D. course and after completing Ph.D. you can join as Assistant professor in any degree college or University.