C182 Flight Instructor reported having to make an off-field landing due to fuel system mismanagement.

Narrative

We had been having problems with the fuel cap on the L wing of this aircraft and were heading to ZZZ from ZZZ1 for repairs. We planned to keep the fuel low since we knew repairs to the tank or tanks was going to take place. We were at ZZZ discussing the cap and several other maintenance issues with the mechanic. Myself and the CFI candidate student became distracted and did not check fuel leaving ZZZ which we should have done since out mission changed. Take off from ZZZ; climb; and cruise were normal. The engine sputtered minutes after we exited Class Delta airspace. We checked fuel tanks; engine kept idling; but would not produce power. We turned north and we landed with a left quartering tailwind on a county road - cross/tail wind was about 12-15 knots from the west/south west and we landed on the dirt road heading about north. This was the closest landing area without wires or obvious obstacles. We did land hard; but safely. We called the mechanics at ZZZ and he brought us fuel - about 28 gallons. We pumped it into the R tank; started up; and took off to the south on the dirt road because the wind had shifted slightly and the road was sloping downward to the south. I flew without the student back to ZZZ and had a normal landing without incident. Had we doubled checked the fuel at ZZZ and refueled; the off airport landing would not have been necessary. Distractions; lack of focus; being in a hurry; and a complete pre-flight could have prevented this issue from happening.