2020:
How do we know this event (the heart attack / gun incident that convinced Bruce to retire - seen above) takes place in 2020?
Because in the second year of Terry McGinnis's reign as Batman he meets the infiltration unit Zeta:
Batman Beyond episode "Zeta": "Zeta was on a long term assignment, infiltrating a terrorist group; six days ago it went renegade." - Agent Bennet to Police Commissioner Barbara Gordon.
Zeta later had his own series:
The Zeta Project episode "Quality Time": "NSA File Review. Subject: Renegade Infiltration Unit Zeta. Unit Zeta went renegade August 5th, 2041, during the infiltration of suspected terrorist group Brother's Day. Zeta is trying to contact Dr. Eli Selig, former head of the Zeta Project and primary architect of Zeta himself. Zeta's objective in contacting Dr. Selig is unknown and prsumed hostile." - National Security Agency computer in response to query by Agent Bennet.
Six Days after August 5, 2041 is August 11, 2041 (a Sunday). This information is what grounded Batman Beyond in the 2040's.
2041 (Year 2 of Batman Beyond) minus 1 = 2040 (Year 1 of Batman Beyond).
This opening flashback to Bruce's retirement is referenced as taking place 20 years before Terry becomes the new Batman:
2040 minus 20 = 2020.
What about Bruce's Age?
In the DCAU there is no mention of Bruce Wayne's true age. But we know that Bruce's parents were killed leaving a showing of the adventure film "The Mark of Zorro" (per JLU episode "For the Man Who Has Everything").
The animated Batman was inspired and shares some continuity with the Tim Burton Batman films, who's look was inspired by graphic novel mini-series The Dark Knight Returns by Frank Miller. In that story the movie that Bruce's family sees is the 1940 "The Mark of Zorro" with Tyrone Power, which was released in November of that year. I decided to keep that homage intact, but to keep Bruce's age in check I theorized that Bruce's parents took him to see the 30th Anniversary of the film in November of 1970.
It has been alluded in several episodes that Bruce was 8 years old when his parents were murdered.
Justice League episode "A Better World, Part II": "...we've made a world where no 8 year old boy will ever lose his parents because of some punk with a gun." - Justice Lord Batman (to Justice League Batman)
JLU episode "Kid Stuff": "I haven't been a kid since I was 8 years old." - Bruce Wayne
JLU episode "Epilogue": "Stop me if you've heard it before. You're 8 years old. Your parents have just taken you to a rousing adventure film; a grand time is had by all. But unknown to you a mysterious figure hides in the shadows. My plan was simple: the killer would leap out at you and kill your family. The trauma would put you on the path to becoming Batman." - Amanda Waller
If Bruce is 8 years old when he sees the 30th Anniversary release of "The Mark of Zorro" in November 1970, then it follows that Bruce would most likely be 58 years old in 2020; 50 years after 1970 (when Bruce was 8); 58 years after 1962 (the year Bruce was most likely born).