B.Sc Chemistry

Yes, you are eligible to pursue M.Sc. Chemistry, as you have completed B.Sc. Chemistry. The basic pre-requisite to pursue a part time course in M.Sc. Is that you should have completed your Bachelors either in Science or in engineering. Few colleges/institutes may require you to appear for a qualifying examination to gain admission.
Here is a list of Universities offering M.Sc. Chemistry through distance learning:
Annamalai University
Alagappa University
Andhra University
Berhampur University
Bharathiar University
Dr BR Ambedkar Open University
Kakatiya University
Madurai Kamaraj University
Maharishi Markandeshwar University
Manonmaniam Sundaranar University of Madras
Madhya Pradesh Bhoj Open University
Nalanda Open University
Nagarjuna University
Periyar University
Sri Venkateswara University
Regards.

Yes, you are eligible to pursue Masters in psychology after completing B.Sc in chemistry honors. Please check for a detailed eligibility criteria mentioned below, along with the admission process.
Eligibility:
Qualification- Graduation
Additional Details
Mandatory Courses : B.A., B.Sc.
50% seats will be filled through B.A./B.Sc. course with psychology as one of
the paper or specialization and remaining 50% seats to any other degree
recognized by Andhra University.
Admission Process
Entrance Test
Eligible candidates need to appear for the Andhra University Common Entrance Test (AUCET).
Counselling
Admission is through counseling based on the rank obtained in Andhra University Common Entrance Test (AUCET)
Regards.

UGC NET 2018 Importance and Summary:
On behalf of University Grants Commission (UGC), Central Board of Secondary Education (CBSE) conducts National Eligibility Test (NET). UGC NET is conducted twice a year in order to select candidates for only Assistant Professor or both Junior Research Fellowship (JRF) and Assistant Professor in Indian universities and colleges. As per media reports, from 2018 onward UGC NET will be conducted once a year.
Please refer to the link provided below for details on eligibility to apply for the UGC NET. As per my knowledge, candidates who have secured 55% and above in their UG degree are allowed to apply for this exam. However, there are other criteria which you must satisfy in order to be eligible to apply. Please check for pre-requisites at the link mentioned below.
https://www.shiksha.com/teaching-education/exams/ugc-net