It is possible to build a Post machine that recognizes the language:
I used to have a brochure "Post Machines", but I never learned the difference between Post and Turing machines. However, according to Wikipedia, they are equivalent. Since the language you've given is Turing-decidable (more precisely, context-free), the answer to your question is yes.
Edit: This probably feels like an insult if you were actually asking for help in constructing the machine... See joke #19 here.