It doesn’t take long for Gilgamesh to determine that Mash, Ana are a waste of his time, as he easily deflects their attacks. He also reveals that the Holy Grail is already among his treasures, which is why the Three Goddess Alliance is attacking Uruk. But as it’s one of his treasures, Gil is unwilling to give it to anyone; not the goddesses (including Ishtar, who makes a brief appearance) and not to Chaldea.
Merlin suggests they stop asking for now; Gil is a moody man, and leaving him alone could bear fruit later. Gil’s attendant Siduri suggests Mash and Ritsuka gain his favor through achievements not in battle, but simply in soaking up the capital and its people, rhythms, and work. If they play ball and show due deference to the king and his city, maye he’ll be more receptive.
To that end, Siduri shows them their modest but adequate new base of operations, where three additional Servants in Benkei, Ushiwakamaru, and Leonidas come to visit, eat, drink, and be merry with Mash, Ritsuka, Merlin and Ana as part of the larger “Uruk Experience.” Siduri also confirms that Enkidu is indeed dead and has been replaced by a fake who answers to the Alliance; but Gilgamesth has yet to meet him in person.
From there Mash, Ritsuka and Ana make themselves useful performing all manner of tasks that while generally menial and perhaps “above” time travelling warriors, are nevertheless tasks that are crucial to Uruk’s survival.
That means not just making mud bricks, harvesting wheat, shearing sheep, and tending to the children and the sick, but also joining Ana in the caverns below Uruk to dispose of evil spirits she believes are contributing to a wasting epidemic among the populace.
Ana doesn’t initially get why Ritsuka and Mash are interacting so closely with that populace, but Ritsuka very logically explains that getting to actually know the human beings he seeks to save helps to motivate him, as well as to more fully empathize with their fate should they fail. And Fake Enkidu and his goddess mother very much want them to fail.