I was mainly just wondering this since there are some characters I want to create, but the classes don't always fit the idea.
This is typically glossed over with the use of skills, tools and backgrounds.
For example, if you had proficiency in Smith's Tools you were likely some sort of craftsman or at least apprenticed to be so eventually. This would be sort of Guild Artisan background (although I don't particularly care for the naming). You get a tool proficiency as part of the background. This would mean that you could be a smith, woodcarver, alchemist, herbalist and so on.
Folk Hero even has some text on this:
You previously pursued a simple profession among the peasantry, perhaps as a farmer, miner, servant, shepherd, woodcutter, or gravedigger.
In order for a PC to actually be a Doctor instead of a Cleric that has proficiency in Healer's Kit you will have to figure out a way to give balanced ability on par with PC classes from the book. Especially, if you have both classes and non-classes in the same group.
This is akin to old NPC classes from Dungeons and Dragons 3.0 and 3.5 (specifically the Expert) though and you could do some work on converting these to work in 5E. In any event you will have to design level progression and balance the abilities, all in all it will take a lot of work. It is worth noting that the NPC classes were less powerful than full PC classes.
This option has the advantage of being able to have this PC advance with the rest of the party, albeit being a bit weaker than the norm.
Something that sort of already exists.
Another option is to use the NPCs from the Monster Manual and add skill proficiencies as needed. These are not really designed to level up and usually are much, much weaker than a standard PC and many would even be non-combatants but not necessarily useless.
Options include starting with Commoner, Guard, Scout or even Spy.
If you create something like this and the result is a character that is somewhat weaker than the system is balanced to handle you will be required to tweak existing encounters in published modules to compensate. Unless, of course you are going for a Cthulhu or Dark Souls style horror/suspense feel.