Yeah! PHD holders are eligible for government jobs exams. Although most of the government exams have age limit of 28 years with 3 years of relaxation of SC/ST. So if you complete your PhD then the age criteria will be expected to be over for most of the exams until and unless  you are from reserved category.

Now the state PSC exams for civil services have 36 year age limit. SO you can obviously apply for that. Apart from this you can apply for College service Commission exam in order to be Assistant proferssors in Government colleges. You can apply for school service commission in order to be Headmistress of Government schools. Also if you are from Physical science background i.e., Physics/Chemistry/Mathematics you can go for different research institutions like ISRO, BARC, DRDO etc. Thus you have a spectra of topics to work on.

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