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When I asked for a prescription at a nearby clinic, they made an openly unhappy face.
Literal
Nearby-medical-place [at-で] prescription [object-を] requested-[when-たら] openly disliked face [object-を] was-done.
A frustrated medical anecdote. 近医 ('nearby medical place,' 'local clinic') is a fairly formal abbreviation — 近 (near) + 医 (medical) = 'nearby doctor/clinic.' The passive 嫌な顔をされた (literally 'had a disliked face done to me') is the adversative/receiving passive: the speaker is the affected party, the doctor is the agent making the face. This adversative passive is a hallmark of Japanese for expressing being on the receiving end of something unpleasant. 露骨に ('openly, overtly, blatantly') intensifies the rudeness — the doctor wasn't even trying to hide it.