Bargarh is not a town where you go looking for a drink in any conventional sense. There are no bars, no pubs, and no cafés aiming at travellers. What the town does have, in abundance, is chai: small stalls on almost every corner serve glasses of milky, sweet, gently spiced tea from around six in the morning until ten at night. The stalls are the social glue of the town — men sit on wooden benches, read newspapers, argue about cricket, and watch the traffic go by. You are welcome to join them; nobody will pay you much attention.
Dhabas — the roadside restaurants that feed truck drivers and bus passengers throughout India — line the national highway at the edge of town and along the road to Sambalpur. The food is standard north-Indian dhaba fare alongside local Odia dishes: dal, sabzi, roti or rice, and often a mutton curry. The dhabas are open early and late, and a full meal costs very little. Hygiene varies; busy dhabas with a high turnover of food are generally safer than quiet ones.
For something resembling a café, look for the South Indian-style coffee shops near the market that serve filter coffee in the traditional metal cup-and-saucer. They are not common, but they exist, and the coffee is better than it has any right to be.