B767 First Officer reported a hydraulic system malfunction during climb which resulted in a gear disagreement EICAS message. The crew returned to the departure airport and landed safely.

Narrative

Gear disagree message/hydraulic leak. I was operating a flight from ZZZ airport. It was decided in our briefing that I would be the PF for the first leg and the Captain would be the PF. After a brief planned taxi delay; due to arrival times and favorable tailwinds; we departed off of runway XXR on the ZZZ departure. We climbed out and cleaned up as per our normal departure profile. Shortly after I made the positive rate gear up" call the Relief Pilot stated we had a GEAR DISAGREE master caution message. We elected to continue the climb and address the message at a less critical phase of flight. Once we reached an intermediate level off I stated "I have the flight controls and radios; can I get a gear disagree checklist please. The Captain and Relief Pilot jointly ran the appropriate checklists; phoned dispatch and Maintenance Control; and notified the FAs (Flight Attendants) and the passengers. We noted as a crew during this time that our hydraulic quantity seemed to be in the normal range. After consulting our manuals and Maintenance Control we came back together as a crew and devised a plan which included leaving the gear down; burning fuel to be below max landing weight; and returning the ZZZ airport. We recalled we had a check pilot dead heading so we had him come up as a fresh pair of eyes to give everything a look over and be sure we had exhausted all avenues. He also agreed so we then notified the FA's and instructed to prepare for a possible evacuation; however we anticipated a normal approach and landing. I notified ATC that we were requesting priority handling and would be returning to ZZZ and requested ARFF (Airport Rescue and Firefighting) to be standing by. Before commencing the approach we noted that the hydraulic quantity had diminished and a RF appeared. At approximately 50ft above the threshold we observed a 10kt increase in airspeed; which coincided with a report from the aircraft in front of us made; and corrections were made for a normal landing inside the touchdown zone. After clearing the runway MX (Maintenance) pinned the gear; advise us to turn off hydraulic pumps; and we were towed to the gate. Upon reaching the gate a ground personnel advised us that our right main gear was smoking and that a substantial amount of fluid was leaking from that area. The jet bridge was already attached and pax (Passengers) had begun to deplane so we elected not to escalate the situation. MX was notified and stated that they were aware. After we debriefed and deplaned we all agreed that we were no longer fit for duty and would not continue to our destination."