Mars (aka Risi Soru the Red Planet) is the fourth planet from the sun in the Solar System.
No action takes place on Mars itself.
According to the backstory the Marine of Doom II stopped an invasion at Mars Base which occurred over the course of a few days. He was the only survivor. He was sent home from Mars on shore leave only to discover that Earth was invaded after his drop pod landed.
Though not mentioned by name, what appears to be Mars Base is visited during the game. It is described to be 'planetary' base twice.
The manual discusses it being location of the classified "Doom Episodes" described as a 'space installation' he once saved, and in Nintendo Power, Our Hero mentions he had returned to the location of his 'first mission' (which technically could be Mars as he was stationed there in Doom 1, but Doom II marine was also stationed there as well). In addition it refers to 'installations' plural in a another seciton which could be a reference to both Phobos and Deimos.
If it is set on Mars, it actually could be a reference to the Doom II's manual's story, which offered a different 'escape' for the Marine (Doom II), after he fought the forces of hell at the Mars Base itself.
Whatever the case, the sky on the "Martian" levels of Doom 64 appears to be simply a starlit background, but the first time you get a chance to get outside the base to see the the Terraformer device, you can make out reddish/brown craggy mountains (shadowed in nightly blue) in the distance (perhaps also indicating the possiblity of it being on Mars). Interstingly if use noclip on indoor levels only stars can be seen in the skybox (meaning that two different skyboxes are used), the stars are only bean to be seen from skylights. Some levels which take place completely indoors and have no skylights have no sky textures at all. Oddly enough the star patterns used in the two sky textures appear ot be what would be seen from earth, with Venus seen in the sky, as well as Orion constellation. While a space installation on Earth was attacked in Doom II, its unlikely that it would be the same location, or that earth would need 'terriforming'.
In Doom 3, the action is set directly on Mars, in the UAC Mars Research Base and Mars City. The player is required to venture onto the barren surface of Mars a number of times during Doom 3, and must fight off demons while topping up their oxygen supply via collectible tanks. Although the planet is portrayed as being realistically inhospitable, a decision was made early on to increase Mars' gravity to match Earth's for gameplay reasons (in reality, Mars has barely 38% of Earth's gravity).
Mars was once inhabited by an advanced civilization that was destroyed by the forces of Hell after creating a teleportation device that led to the opening of a portal into the netherworld. The Martian race, in desperation, sacrificed their remaining population to create the Soul Cube which was then used by their champion in defeating the demons and sealing the portal to Hell. The few survivors left Mars and are implied to have traveled to Earth, where they became ancestors of modern humankind.
The backstory treats it as a sequel to Doom 1-3 (or at least in the same timeline, the exact order is not clear), it is set in the Mars Outpost the oldest facility on Mars, older than the Mars City.
In Doom II RPG the main characters (a different set of characters from the character in the first game) transfer from Mars to go to the Lunar Outpost.
Deimos is missing from Mars skies which seems to be a nod to the events of the original Doom/Ultimate Doom. However, its orbit can be seen, so its possible its still there, just not viewable on screen.
Note:Mars in Doom (2016) might be the same mars in Doom 3, set four years later, as they share much of the same timeline, see shared timeline.
Doom Slayer can now breathe on Mars (he is technically wearing a full self-contained Praetor Suit however which handles all life functions), which makes it easier to fight monsters. In-game datapads also suggest that the UAC has performed some level of terraforming on the surface to make the atmosphere more breathable. The campaign begins at Mars Installation in the Hellas Plain, then Argent Facility, and then the UAC Advanced Research Facility. Other locations mentioned include Meridiani Spaceport. There is an offworld orbital research station above Mars called the UACSS Picus.
Mars reappears in Doom Eternal, where it is revealed that the planet used to be habbitable, and was colonized by the Bethian clansmen from Argent D'Nur who built the Sentinel City of Hebeth on its surface. Hebeth was renowned for its slipgates that connected the Sentinel Worlds together, but it, along with the rest of the city, was lost at an uncertain time when Mars was rendered uninhabitable and Hebeth sunk to Mars's core.
Despite this, the slipgate to Sentinel Prime still functioned well after the city collapsed. This gate became crucial to the Doom Slayer's efforts, as he needed to reach Sentinel Prime where Deag Grav was located. Since there was no other way to reach Hebeth, the Slayer utilized the BFG-10000 and fired it at Mars, breaching the crust and exposing the planet's core.
Mars continued to break up well after the BFG was fired, and though there are no mentions of its current status, it is presumed that the planet broke apart.
Behind the scenes
Mars appears in Doom 4, it is set directly on Mars as in Doom 3, as the two games are possibly connected (at least both share Soul Cube and Super Turbo Turkey Puncher 3 machines. Although the UAC corporations in both do not seem to share any of the same history (see Doom 3 timeline and Doom Reboot timeline), and have vastly different backgrounds. It is more likely that Doom 2016 is set in a parallel universe with some similarities to Doom 3 Mars, but different. UAC in reboot universe has a cult like structure that worships Hell, and many want it to invade the universe, while Doom 3 they were just your standard military/corporate types experimenting with new technology who unwittingly unleashed Hell.