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I am playing on a randomly generated map, and there doesn't appear to be any oil reserves anywhere nearby. Is that possible, or do I need to travel far and wide to find oil?

Depends on your map settings, default settings will definitely spawn oil in reachable distance.

On default settings, there is usually an oil patch within a Radar's reveal range of where you spawned. If you have a Radar down and the revealed map is a square and doesn't change from that in ~30 seconds, then your Radar has surveyed as much of the map as it will. If you need it ASAP, doing some manual scouting can help, though you made need to plow through some of the nearby biter bases to reach it. Not too hard if you get yourself some Heavy armor, SMG (perhaps Shotgun) with plenty of ammo (mags are 10 shots each, so, especially with the shotgun, you don't need a ton to last a while, SMG can chew through mags pretty quick), and the pro tip for early combat: go search nearby lakes for little black things that move, and right click those till you pick em up (have to chase em a little, but you can reach them from very far away), those are Raw fish, you can consume them in combat to heal (80 health each time, your max is 250 for comparison), which is amazingly helpful in harder early game fights. With that you should have no trouble reaching nearby oil, in a default settings game at least.

Oil is usually a little further away (EDIT: Compared to the four starting resources). Keep searching. To actively look the quickest way is the car, which will also outrun any biters that pursue you. Just watch out for cliffs, rocks, and trees. Radars can passively reveal terrain over time.