Estimating the target location for splashdown from the various NASA space missions was a critical concern; the first test of Mercury ("Big Joe") landed 760 km short and between two of the backup destroyers on the flight path. After landing it wasn't found for three hours (they used search aircraft to find the locator beacon). (The "flight path" implies something to keep in mind in all this -- the actual path of orbit was known fairly exactly, the problem was undershooting or overshooting along that path.)

Needless to say, that was not an acceptable recovery time; the longest recovery of a manned vehicle was Gemini 5, which missed the target by 170 km and had a recovery in 1.5 hours.

This wasn't the bigget manned miss -- as implied earlier, there were backup ships to account for landing too short or too far. Scott Carpenter of Mercury 7 missed by 390 km but only had to wait an hour for rescue.

The closest splashdown to a ship was Gemini 9, at 600 meters (about a third of a mile, a little too close).

The ability to project a landing spot improved with practice; while Gemini started with 14 secondary ships that number was eventually reduced to 8. Gemini 10 landed 6.2 km from target, Gemini 11 at 5 km, and Gemini 12 at 5 km again.

Apollo, being a different vehicle, took some adjustment; they missed by 72 km and 370 km respectively on the first two (uncrewed) tests. By Apollo 7, they started being fairly consistent:

7: 13 km

8: 5 km

9: less than 2 km

10: 5.5 km

11: 24 km

12: 6.9 km

13: 6.5 km

6.5 km is about 4 miles. This means the image you linked is fairly reasonable (although yes, a little exaggerated).

...

Dirk H.R. Spennemann (2005) The naval heritage of the US space programme: A case of losses, Journal for Maritime Research, 7:1, 170-214, DOI: 10.1080/21533369.2005.9668350

It's also worth pointing out that this wasn't simply a job of estimating where the spacecraft would land, as though the spacecraft was just a tossed rock that a catcher on the ground had to position himself to intercept. The spacecraft had its own means of controlling its flight path, and its abilities improved through Mercury, Gemini, and Apollo.

Mercury was a ballistic spacecraft, meaning - like the name implies - that once it fired its retrorockets, it was flying on a ballistic arc like an artillery shell, without any means of controlling itself. The only control the pilot had over his impact point was timing the exact moment that he fired the rockets, and orienting the spacecraft as precisely as possible so that the rockets were facing the right way. There was a mechanical clock in the center of the instrument panel that counted down to retrofire, and this could be adjusted by the astronaut in flight as computers on the ground refined the spacecraft's trajectory. Once the rockets were fired, you were going where you were going. Scott Carpenter on MA-7 missed by so much because he fired his rockets late, and was slightly misaligned when he did it.

Gemini could do much better, having the first digital computer onboard a manned spacecraft. The Gemini spacecraft also had an offset center of mass, so that the astronauts could actually "fly" it during reentry. The computer was programmed with the intended landing point and would drive attitude needles on the instrument panel. The astronauts would follow these needles, and by rotating the capsule along its long axis they would alter the location of the capsule's center of mass and thereby the flight path, letting them steer left, right, uprange or downrange. This computer guidance would let them get within a 10-mile circle. As the examples provided above showed, this worked very well in practice.

Apollo could do even better, having a similar offset center of mass and a much more capable computer. During reentry, the computer would run Programs 63, 64, 65, and 67, known as P63-P67. These would automatically fly the spacecraft using a similar method as Gemini, with the added ability to partially skip out of the atmosphere to lengthen the flight path, if necessary. With this improved capability, all of the manned Apollo flights landed 3 nautical miles or less from their intended aim points (not from the ship, mind you, but their aim points).

Sources:

Orloff, Richard W. Apollo by the Numbers: A Statistical Reference

Shayler, David J. Gemini Flies: Unmanned Flights and the First Manned Mission

Woods, W. David. How Apollo Flew to the Moon