Hungarians have a weird obsession with gyro. To appreciate this, all you need to do keep your eyes open as you roam downtown's streets, which are teeming with gyro joints. These places generally serve cheap food but it's made from low-quality ingredients and usually doesn't taste very good.
Gyros Kerkyra is one of the few shining stars in Budapest's dismal gyro universe. While the place also serves bifteki (ground veal patty) and souvlaki (grilled chicken and pork skewers), the classic chicken gyro is the one to go for. It comes with sliced chicken fresh off the rotisserie, tzatziki, spicy feta sauce, creamy carrot salad, and some fries, all inside an original Greek pita (€3). Beer and wine are also available.
Try to visit Gyros Kerkyra outside of peak lunch or dinner hours to avoid the lines, which can stretch outside the building. If you have time to discover the neighborhood, you might find this guide of District 6 helpful.