Uh, with shovels. Seriously, most modern, rural wells are drilled, giving you just enough shaft diameter to drop in the neccessary piping. Sometimes wells are bored, which I think can only be used for shallower wells, but give you a larger diameter hole (also, more water reservoir per foot of depth). Required depth depends on finding an underground “stream” of water.
But the house my father built some 40-50 years ago has a well that he hand-dug by himself. The well shaft was about 3.5 - 4 feet in diameter, which was the diameter he needed to comfortable use a shovel and mattock. He dug/cut handholds into the side of the shaft which he used to climb in and out and he had my mother and/or a friend of his stay up top and wench the buckets of loose dirt out and send the empty bucket back down to him. The well was only about 50 feet deep.
How long, I would guess a few days of dedicated work by a couple of diggers, depending on soil conditions – which, just let me say, I sure this is a very dangerous thing to do, don’t attempt at home, etc, etc.
If you’ve got access to the old Dungeoneers Survival Guide book, there are charts in there for how much earth can be moved by various races per day in a mining situation, use that as a basis and modify if you think necessary. I bet digging vertically is at least half as slow as digging a horizontal shaft – maybe slower.
And just like modern wells, when you get down to where you strike water, they usually insert some sort of material (concrete sleeves for modern wells / stacked stone walls for ancient ones) at the bottom to keep the water from erroding the walls and causing a cave in.