My son cried when Yoda died. He asked me why no one took care of him during his last days. Given that Yoda was around 900 years old, surely he would have lots of descendants to look after him. Why were his offspring so unfilial?
Yoda has no known family; his background is deliberately kept vague.
From an interview with George Lucas:
So he's a mystery character, he's a magical character. He has no background. He comes and he goes. He's the subversive secret mysterious stranger that enters the film and then exits at the end.
Yoda's page at the Star Wars Databank lists his species as "unknown" and makes no mention of any family. In this answer, @Richard lists all members of this species to have appeared in any licensed Star Wars media. It's a short list (Minch, Oteg, Vandar, Yaddle and Yoda), and none of them are Yoda's kids.
And this page, claiming that "the Children of Yoda and Yaddle were a bunch of weird asshole frogs", is decidedly uncanonical!
Yoda was a Jedi, which put restrictions on his life choices.
First, Yoda joined the Jedi Order as a youngling, which meant he was a Jedi since he was a child. It is normal Jedi practice to take very young children from their families to begin Jedi training as soon as possible. This is why Yoda initially opposed Anakin Skywalker's training as a Jedi, since Anakin was not discovered by the Jedi until he was nine years old. Anakin was still a boy, but the Jedi considered him too old to begin training!
Also, the Jedi forbid marriage because the Jedi are required to avoid attachments and possessions (this is why Anakin and Padme had to hide their marriage). Technically, it's permissible for Jedi to have casual sex so it would have been permissible for Yoda engage in sexual relations with a member of his species or of a species with which he could reproduce. That said, it would be extremely difficult to not become attached to your children so the Jedi would discourage having children as well.
In short, Yoda was a Jedi all of his adult life and the Jedi forbid attachments, so Yoda would not have been permitted to have a wife or children.
I'm going to be the devil's advocate here and assume Yoda had a family on the side that he never told anyone about. Maybe some six-inch tall kids, a little green wife, and a cozy home on some distant moon. Let's assume this is true.
Now, look at the events of Episode III. The entire army of the Galactic Republic has standing orders to kill all Jedi, at all costs. Right before the vast majority of the Jedi were wiped out, a small team of them tried to assassinate the Emperor; after this failed, the Emperor told the story like the Jedi were trying to overthrow the government.
So, essentially, Yoda is a terrorist, probably public enemy #1 in the entire galaxy. The last thing he'd want to do is go anywhere near his family; for one thing, anyone who knew about them would be expecting it, for another if he got caught, it's very possible his family would get killed in the crossfire. The best thing Yoda could do for his family is make them think he had died on Coruscant; they would mourn his death, but they would be safe.
If Palpatine had not destroyed the Jedi, the Order would have taken care of Yoda in his old age, just as real-world monastic orders take care of their elders. So, before the destruction of the Jedi, this would not have been a worry for him. And after the destruction of the Jedi, Yoda could not return to his own kind, because he could not leave Dagobah.
There are plenty of old people, living in the real world, who have children that don't take care of them. So it's faulty logic to say that because no children are taking care of him, that he has no children. He also may have had children who failed to outlive him.
In the books of the Thrawn Trilogy another reason is given...
Remember that weird cave that Luke entered in Dagobah? He fought Darth Vader in there, or at least, a being that looked like him that turned out to be Luke himself.
This cave is actually embedded with the dark-side and that fight was a lucid dream Luke had (created by The Force).
Why does this relates to Yoda?
This cave can be considered as a dark-side nexus, which may be created when a strong dark-side user dies.
When Yoda was seeking a place to hide after the events following order 66, he may have felt the presence of this powerful dark-side user and followed his trail to Dagobah. Yoda eventually killed him in there.
Since Yoda's light-side power was too great to hide from the Emperor's perception and perhaps other force-sensitive beings from the Empire itself, he chose specifically to continue living in Dagobah, masking his light-side power with dark force emanating from that place. I.e. a light-dark-force balance was reached.
Relating to the children's question...
I suppose a being of his age could have children, but one hypothesis given in the books is that Yoda was the last being from his own species (we have never seen any other being that looks like him and no mention to his species was ever given). Adding to that, if he somehow had children, consider that they would most likely have been force-sensitive as well and could have been tracked down because of that....
Yoda is a hermit, and hermits live mostly alone. Obi-Wan lived the same kind of life on Tattoine until Episode IV events. Yoda just lived it more extremely, living alone in a whole planet.