I know you asked for a place to eat and then a place nearby to drink but two places come to mind when I hear "good reasonable bar food", Dublins and Glascotts. Dublins is in the gold coast and when it comes to bar food at a reasonable price, I think it takes the belt. They seem to do most of their dish's to an above average rating and the service is always good, when your done eating, start drinking without leaving the establishment.
Any of you that have been to Glascott's might be saying "huh", they dont serve food" and no they dont, but for years now Glascott's has paired with the adjoining, Athenian Room. Glascott's provides diners at the AR with drinks and the AR supplies drinkers at Glascotts with food. As far as bar food goes, I dont see how you could ask for anything better, gyros, burgers, chicken, Greek fries and the skirt steak all really hit the spot and the skirt steak is the priciest at $12. This might be exactly what you are looking for, good "budget" food (AR) and a bar that you dont even have to step outside to enter in Glascott's next door.
Dublin's
1050 N State St, Chicago 60610
Btwn W Oak & W Maple St
312-266-6340
Athenian Room
807 W Webster Ave
Chicago, IL 60614
(773) 348-5155
Glascott's
2158 N. Halsted St., Chicago, IL 60614
Tel: (773) 281-1205