Air carrier Captain reported that after long departure delays due to heavy snow and de-icing; the landing gear would not retract during takeoff. The flight returned for maintenance and crew rest however after planning for another departure the crew reported too fatigued to fly.

Narrative

Scheduled to operate Aircraft X. Got to the gateway on time XA30 pm and there was light rain on the field. By the time we were loaded up the rain changed to a very heavy snow. The weather was calling light snow -3C. The snowfall intensity table entering those conditions showed it to be moderate snow and gave me a holdover time of 30 minutes to an hour. I called my Dispatcher and he agrees. We then pushed back to taxi to the de ice station. During taxi the snow had already piled up and the breaking action became nil. I slid to a stop and set the brake and reported to Ground. They sent snow plows to me and cleared me a path back to the parking spot. It was just coming down too hard and I figured the runway would be bad also. I figured that even though the weather was calling light snow; in my judgement it was actually heavy snow with a rapid rate of accumulation. We made it back to the gate and waited 4 plus hours for the conditions to improve. And to allow them to plow the field.During this time I worked with Dispatch and also informed Operations of our situation. The airport was putting out very different runway condition reports so it was hard to determine safe and legal. Reports were rapidly going from 3/3/3; 1/1/1; 6/6/6. There were 4 other flights dealing with the same issues as me. The Company flights cancelled at around XC00 pm and us and the 2 cargo Jet planes sat at the ramp waiting for the conditions to get better. After the long wait the snow let up a bit and all surfaces had been plowed so I decided to try again. I had the numbers I needed. One of the cargo jet had already taxied to the de ice pad and I asked him how the taxiway was and he reported them as fine but they were returning to the gate because their snowfall intensity table was more restricted than mine they needed 1 mile visibility for the weather to be considered moderate snow I only needed 3/4 of a mile. I then taxied to the de ice station. It took about 1.5 hours to de ice. The snow never stopped but was going back and forth from heavy to light.After we were de iced we were cleared for takeoff and the visibility was 1.5 miles and the runway was plowed. After liftoff we couldn't retract the gear. Due to the icing conditions; I was now in an urgent condition and needed to get on the ground fast. We [advised ATC]; asked for immediate vectors back to Runway XX ILS Cat 2. The weather app on my EFB showed tops to 15000 to 25000 ft. and the storm went for hundreds of miles around me so where I took off from was my best bet to get out of the ice. We threw in the approach; quick brief; ran all checklist; checked for any checklist guidance on the gear but found none. We tried twice to get them up but only the doors would open and I didn't want to risk getting them into an unlocked condition. The Controller gave me short vectors to about a 6 mile final. I put both autopilots on; selected med auto brakes and we couldn't get the ACARS to provide landing numbers so I flew hook plus 10. 5 for ice and 5 for gusty winds. Somewhere around 500 ft. we saw the runway. I allowed the autopilot to get us to 200 ft. and turned them off. I didn't know how it would perform in those conditions The landing was uneventful and I sent the CFR guys home. We then were told we could not park back at our gate and had to wait a few hours for them to plow remote parking for us. Once in remote parking we waited for stairs and a ground crew. Shut down. Called Company. Plan was for us to try again as we had 2 hours duty left. After what we just went through I determined that to be unsafe and said no. They coded us as Fatigued and called the hotel. We had to wait for an hour or so for ground transport due to the bad road conditions. Got to the hotel at XK30 am with a plan of leaving out again at around XD00 pm. I got in bed at XM00 am and fell asleep. At XQ37 the phone in my room rang. It was the front desk saying to call Scheduling. I called them and was informed that we now had a much earlier departure. XU00pm van. I told him he disturbed my rest; I've only been sleeping for 4.5 hours after a long stressful duty day. The tone in the Schedulers voice changed and I could feel the pressure amping up. I asked if my FO (First Officer) had also been called and he said yes and he was ok to fly. I told him I'd call the FO and ask him myself. My FO said he was ok so we agreed to have a XU00pm van and try to get out of there again. We figured we'd show up; step on the jet and fly home. Arriving at the gateway we took them by total surprise. There was only a few office staff there. Told us it would be hours before workers showed up. I decided to use that time to get with Dispatch as the jet would be flown to ZZZ1 gear down. That was an extensive flight planning session and it was determined that we could only weigh 262000 in flight in case I lost an engine and needed to stay above the clouds as you can't fly in icing conditions with the gear down and there were clouds enroute with tops to 10000 ft. We also had to multiply our fuel burn by 2.2 per the charts. In order to do that the 60k of cargo would have to be unloaded. There was nobody to do that. At that point I was so tired I was struggling with even simple tasks and decided the mission was too complex and would take too long for my crew to safely fly. I informed the dispatcher; and 2 [other pilots] that we were too tired to fly and they all agreed we were not rested enough for the flight. We were again coded as fatigued and sent back to crew rest. During that rest we went from laying over for 24 Plus hours to being jump seated home the next morning.