Dr. Aaron Smith, Ph.D. in Educational Leadership, Currently Program Director at Aviation Academy, Co-Author of Awakening Your STEM School
It is quite possible, however there is more to getting a job than just course work. We partner with NASA-Langley and have learned from multiple employees within the facility that work experience must also accompany coursework.
If you are serious about going to NASA, I highly recommend that you look into any internships or Co-ops that they provide from time to time. Go to this website to find more information.