Here are some of the requirements for each of the Sagas, with some requirements still to be announced in the future:
Character Requirements:

Pestilence Saga - Morgan Le Fay and four Darkhold characters (Agatha Harkness, Wong, Doctor Strange (Heartless), and Scarlet Witch)

Famine Saga - Rogue and four Unlimited X-Men characters (Gambit, Fantomex, Sunfire, and Dazzler)

War Saga - War Horseman (TBA) and four other characters (TBA)

Death Saga - Death Horseman (TBA) and four other characters (TBA)

Apocalypse Saga - All Four Horsemen, their respective teams, and several additional characters

For the four Horsemen Sagas specifically, these are the Gear Tier requirements
Gear Tier Requirements:

Difficulty 1 (Hard) - Required Gear Tier 13 for each character
Difficulty 2 (Heroic) - Required Gear Tier 15 for each character
Difficulty 3 (Apocalyptic) - Required Gear Tier 17 for each character

The fifth and final Apocalypse Saga will have another layer of depth with different gear and Iso-8 requirements, which will be announced in the future. Additionally, to enter the final Apocalypse Saga, you must have completed all of the four Horsemen Sagas at least on Difficulty 1. To participate in the Apocalypse Saga on Difficulty 2, you must first complete all of the four Horsemen Sagas on Difficulty 2. To participate in the Apocalypse Saga on Difficulty 3, you must first complete all of the four Horsemen Sagas on Difficulty 3. To first unlock Apocalypse, you must complete Difficulty 2, and to unlock the rest of his Yellow Stars and Red Stars you must complete Difficulty 3:

Difficulty 2 (Heroic) - Gear Tier 16, Iso-8 Blue Level 4
Difficulty 3 (Apocalyptic) - Gear Tier 17, Iso-8 Blue Level 5