The Story of Sans, Papyrus, and Gaster from what information I have on the situation, and some guess work.

What do we know about the font style characters?

1. They are dead/undead/Feigning death
2. Two of them are confirmed to be close to each other
3. All of them show an interest in Science/Mystery
4. Papyrus acts like a child with large aspirations, and an almost inability to deal with complex emotions such as love
5. Sans is intelligent, self aware, and a corny "cool dude"
6. Sans is in possession of an badge, and in possession of a device of unknown origin and intent
7. While fighting during a Genocide run, Sans will have flashbacks thru dialogue about space anomalies in which time is disrupted (almost as a First Officer would tell a Captain a status report)
8. To spite being "dead": Sans can bleed
9. Sans is aware of the universe ending if this time continuum anomaly continues

With these facts, I hope through my next few deductions you will be able to deduce the same as I have.

To be talking/knowledgeable about Space anomalies, and for them to have the instruments to read such anomalies (or through their vernacular), then they must have been The Crew of a Science Vessel. (This may also be deduced from the badge, and the unknown object)

A acute possibility is that this Science Vessel was a family endeavor/Residence vessel/Housed families of the Scientists on board. This may be an explanation for Papyrus's young mind, if he was a minor resident of the Vessel, when the anomaly occurred. Also from this we may deduce that Sans was a Bridge Officer of some capacity with him reading their sensor feed.

From here the story diverges into two foreseeable ways:

1. This anomaly almost undoubtedly began to encompass the vessel, and a hull breach caused mass death. Those lucky enough to survive the anomaly found themselves in a world they did not understand, and they began to change into creatures of different varieties from exposure to undiscovered forces.

2. The Captain of the Vessel, whom I believe is Gaster, in an attempt to save himself, the children, and/or the officers on board began a emergency evacuation by conventional/unconventional means. Causing the people who evacuated to be poisoned by temporal forces / teleportation in extreme cosmic conditions.

If the survivors of this anomaly are Sans, Papyrus, and Gaster. It would not only make sense that he was trying to save his (font) family, but they were all exposed to a(n) dangerous radiation poisoning / amount of Exposure. Without knowing the nature of this anomaly, or how it affected those whom were exposed to it, I cannot deduce the probability of if they would be able to survive it; I also can't tell if they would be able to survive as entities without surviving as organisms.

Gaster's last ditch effort to save his family would explain how they "Just appeared one day".

Also explained in this scenario is as to how Sans has seen the sun, and why he has memories of the cosmic distortion.

That sentence is vague, so I will make a point to say I believe that Papyrus was born in space, and very young when he died. I would even say he was barely in Juvenile Gestation. It is a possibility that while in exposure to time distortion his body was aged beyond his mind, and this accounts for his child like tendencies.

Sans has a knowledge of what is happening to time, and space. He breaks the 4th Wall with leisure, and is confirmed to be able to retain memory once someone has reset the timeline.

It is quite possible that from exposure to the cosmic anomaly that Sans gained not only the ability to change form, but the ability to change physics. Demonstrated by the fight with him during the Genocide Run. Note: The full extent of his abilities were probably dormant, unless provoked by emotional trauma.

What happened after this is, Gaster went on to find a Power Source with enough energy to power a vessel that would suit the needs to find this anomaly, and return himself with is family to their time. Sans began to build a vessel with the abilities they would need to search the Planet / Solar System / Galaxy. After realizing that in this Time, and especially in a world locked away from Modern Civilization it would be impossible to find a pre-existing source of power for his machine, he decided to become the "Royal Scientist". With the authority that accompanied this power, he was able to build his power source.

This feat was the thing that made it once again a possibility for the (font) family to return home... However, Flowey (the initial cause of the disruption) had other plans... This was the end of Gaster's story...

With a power source, and a machine, what would have been a simple way for Gaster to reach space, turned into an impossible task for his son (Sans) to accomplish.

Realizing this... Sans believed that he would be doomed to be stuck there forever, and he just allowed himself to find comfort in those around him: those who he was orphaned to, those who cared little for him, those he believed to be his friends.

I believe that Sans was able to feel the disruptions in the time space continuum, but never had the emotional drive to activate the powers within him to change it. At the loss of his brother Papyrus... Sans was emotionally awakened. He fought to try to protect the world that he was from (Earth), and maybe end the disruption to the space time in order to save his planet from being destroyed by anomalies like the one that destroyed his Vessel.

Alternatively if you choose the Pacifist run or Neutral, then Sans does not need to save the world, and he may live on as Sans the Skeleton, and with Papyrus The Royal Guard.

In conclusion: Sans is a new type of being that is an Entity rather than an Conventional Organism; Gaster was a father; Papyrus was a younger brother aged by the power of a cosmic entity; Flowey killed Gaster in an attempt to prevent Gaster from finding the ruptures created by his ability revert time, and finding out how to open / close / use them; You are either the new abuser of the Cosmic power to revert time, The last person to use the power, or the original being that created the distortion.