Yeah! You can definitely get government jobs. Now as you have BEd you can apply for Primary school teaching. For Secondary and Higher secondary school an MSc is must. Now apart from this you can apply for any exam which have a criteria for graduate candidates in State PSC. Apart from this you can go for SSC CGL, UPSC CSE etc kind of exams to be an officer of the Central Government. SO you have a lot of scopes ahead and based on your interest you just have to choose the right path for you.

For Government Job exams in India visit the link provided below:

https://competition.careers360.com/articles/sarkari-exams

I hope this answer helps. All the very best for your future endeavors!