The closest medical facility to BME is GYÓGYÍR Egészségügyi Szolgáltató Nonprofit Kft.
1117 Budapest Fehérvári út 12.

If you feel sick, first please call one of these doctors on weekdays from 8:00 to 20:00, they will guide you what you need to do:  
Dr. Horváth Péter  or Dr. Szekeres Orsolya available on the following mobile phone numbers: (+36) 30/678-6450, (+36) 30/815-2218. Both of them speak English.

If you need a counseling psychologist, visit this page for more more info:

Cost of treatment:
Citiziens of the European Union:
This case you should have an EU health insurance card. EU health insurance covers most of the charges for medical treatments excluding cosmetic, dental treatments and a couple others. It is advised to ask the doctor or medical facility if you are not sure. 
More information here:

Everyone else:
If you have a health insurance, you should see about what kinds of treatment it covers. Also ask the medical facility if they accept your insurance. 
Otherwise you have to pay for the treatment. This is highly unlikely as a valid health insurance is obligatory for all international students in Hungary.