1. 'Your age should not act as a preclusion to your being accepted on the university course.' 2. 'Your age should not act as a preclusion to you being accepted on the university course.'
In #1, 'being accepted' is a verbal noun, and so can take a possessive determiner. However, this construction
I accept "Despite my being thirsty I went on working", yes. Leaving out the "my" is of course possible and perhaps even preferable because of the "I" which appears quite soon after.
Let's change the sentence so that the conjunction of "me" and "I" is not at issue.
I accept "Despite my being thirsty, they refused to give me water".