Hercules: The Legendary Journeys (1995–1999)
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A good question. The first explanation is that the Olympian gods are capable of using their magic to time travel as has been proven in 'You are there' and 'Armegedon Now'. Therefore Strife travelled forwards in time with Ares into the 1990s and then returned to ancient Greece where he was killed by Callisto later in his personal timeline. This poses the problem of Ares potentially knowing the future which he shows no sign off but his ability to time travel may be limited solely to attempting to wreck Hercules' TV show or a higher power such as The Fates, Zeus or Hera prevent him from learning anything in the future that would upset destiny in the past. Second explanation is that Ares has somehow brought Strife back to life in the present day. Whilst characters who have died return from the dead all the time on 'Xena' and 'Hercules' for the most part gods don't, they stay dead (in fact Valesca aside unlike mortals of all religions it's unclear if deities even have an afterlife at all). It also seems unlikely that if Ares could bring another god back to life he would pick his nephew Strife whom he obviously despises, his twin (and occasional lover) Discord seems a far more likely candidate, not least as it's hinted they're also lovers. Third explanation is that by killing Strife Callisto actually changes history. When we see the 'Hercules' cast again in the later story 'For those of us just joining us' everything is the same EXCEPT that Strife no longer accompanies Ares, nor does he show up in Ares' two later appearances in the modern day on 'Xena'. Possibly the alteration in the timeline was engineered by Dahak as part of his world conquering scheme and was later corrected by the God of Eli or some other protaganist. Fourth theory is that the writers screwed up.
(OP: Discord was Ares' twin, not his niece. Just needed to fix that :) )
(Edited: Whereas it is true that the question is not really answered in the series, some things mentiones above should be corrected:
1. Ares does not at all despise his nephew Strife, he is just often annoyed by his impudence. 2. Discord is not necessarily Ares' twin; she was in the original myths, but since the series changed/altered many of those, it is only safe to say that she is his sister. 3. In the modern day timeline, Hercules interacts with Ares and Strife in a way which suggests that he meets them on a pretty regular basis - like in the old times -, which eliminates the possibility of Ares and Strife meeting modern day-Hercules only because of a time travel. Also, Hercules explicitly says that Ares and he have been fighting for about five thousand years now. 4. In the modern-day episode, Ares and Strife mention Zeus not liking actors and mockingly ask what he thinks of the show, which means that Zeus is still alive - although he, like Strife, was "killed" in the old days. Again though this may be the result of events later being altered in the timeline. 5. In modern day, Ares tries to pinch merchandising rights from Hercules as a "compensation for the weekly slander he has to endure".
Combining all of this, I think the most likely explanation for the "Strife still alive??"-phenomenon is that the series EXCEPT for the modern-time episodes is just the series that Hercules aka Kevin Sorbo is producing. Which means that what the "old-times" episodes show us does not necessarily have to do with whatever truly happened. The death of the gods - probably as well as many other things - was just something Hercules made up for the series... maybe in order to prevent the humans from believing that the gods are still among them, which they obviously are but lacking the powers we've seen before.) However this goes against what we see later with Kevin Sorbo/Hercules insisting on historical accuracy in the show. It also begs the question that if the Greek gods survived why would they allow the rise of Christianity/Islam/Buddhism/Judaism etc as the world's dominant religions?
Simplest answer is we must accept the modern world as it is depicted in later episodes of Xena/Hercules and presume one of these solutions retroactively explains the Strife paradox. Edit (Coming Soon)
Actor Kevin Sorbo suffered a minor stroke forcing him to drop out of filming whilst he recovered. As a result the writers were forced to improvise storylines with either no Hercules in them at all or the bare minimum, writing stories concentrating on other characters or which flashed back to young Hercules (played by a different actor) and even such conceits as transforming Hercules into a pig. They even produced an episode concentrating on the writers of the Hercules TV show forced to improvise new storylines after the loss of their leading man. Edit (Coming Soon)
Ignoring clones and reincarnations;
Hercules=still alive in the modern era and masquerading as actor Kevin Sorbo.
Iolaus=lived to 100 and died of natural causes.
Ares=still alive in the modern era but significantly depowered.
Autolycus=reunited with his wife and opened a bank.
Salmoneus=last heard off thrown in jail for tax evasion. Edit (Coming Soon)