A320 pilot reported a hydraulic system failure during initial climb. Flight crew continued to destination; performed a manual gear extension; and landed safely.

Narrative

Shortly after take off out of ZZZ; we experienced a HYD G RSVR LO LEVEL ECAM. I complied with and accomplished all of the ECAM items as well as the QRH procedures up to but not including the manual extension of the landing gear. I then contacted dispatch via SATCOM and included Maintenance Control and the chief pilot on duty to advise them of the situation. After a brief discussion and input from all involved; I decided the best course of action was to continue to the destination. I then briefed the number one flight attendant regarding the issue and gave her a briefing. As we got closer to ZZZ1; I briefed the passengers regarding the issue and told them they would see emergency equipment on the runway and that I would be stopping on the runway so that we could be towed to the gate due to not having nose wheel steering.Once all those items were complete; I notified ATC and advised them of the situation. I [requested priority handling] and briefed them on requiring a longer downwind so that I would have time to manually extend the landing gear. Once on downwind for ILS XXR; I accomplished the manual extension procedure in the QRH. Once procedure was accomplished and completed; an uneventful landing to XXR occurred. We were met by CFR (crash fire rescue) and Maintenance with a tow crew. The aircraft had an inoperable APU and the tow crew required both engines shut down for the tow back to gate XX. I briefed the inflight crew and the passengers and proceeded to get towed into the gate.The specifics of the events are as follows. Right after selecting gear up; I received a momentary master caution chime along with seeing a HYD G message. The chime and ECAM disappeared. After selecting flaps zero; I received the same momentary master caution which also disappeared immediately. With no active ECAM; I transferred controls to the FO and looked at the synoptic hydraulic page. I saw amber hydraulic quantity along with 3050 PSI in the Green system. Even though it looked like there was no green hydraulic quality; there was no ECAM present. Shortly after passing 3000 feet; the ECAM for HYD G LOW LEVEL latched on to the system. I performed all required items on the ECAM and QRH and contacted dispatch and asked him to include Maintenance Control and duty pilot. I reviewed the additives for failure and crossed checked them with dispatch and duty pilot. We needed to add 1.2 to the max manual breaks distance which kept us well below the available runway for XXR. As a point of discussion with the duty pilot; he recommended I use the failure code associated with the HYDG failure in the performance section of the manual. I ran those numbers with the failure code entered into the perf weight and balance page for landing numbers. The numbers came back identical to the normal numbers. I did the math and applied the 1.2 additive which confirmed the required distance. The next day I discussed this code with my check airman; he advised me the codes are not yet active and that's why the landing distance numbers came back the same.Once I was abeam the airport; I proceeded to finish the QRH procedure for manually extending the gear. The gear manually came down as advertised however; it only took two turns of the lever vice three. I used two hands and could not get past the detent to get the third full turn. After arriving at the gate; the appropriate log book entries were made.