The horizon is pretty much flat when observed from pretty low. It is not a perfect straight line. In fact, what you observe is a circle with several kilometers of radius that has its centre only about four or five meters below your eyes (namely the tangent circle of a cone and the earth sphere where the cone apex is in your eye). That is almost as if you observed that circle from within its plane, in which case you would observe a straight line exactly. More importantly, if you rotate your head left or right, you will not follow some further downfall of the curved horizon line, but instead its central point directly ahead of you will again be highest at (pretty close to) eye level. Unless you adjust your direction of sight: In order to look exactly at the horizon in front of you, you'd have to nod forward by a very, very tiny angle; if you then rotate around the tilted, not quite vertical axis of your head, you'd look slightly above the horizon after half a round.
Eyes are unreliable instruments anyway, so let's build a camera obscura from a used shoe box. The horizon on the image plane on the back is very close to a straight line and is only approximately at eye level, i.e., with a properly aligned box, the horizon will be slightly above (inverted image!) the middle of the screen and the outer ends will be even further above. Now if you rotate the box around the vertical axis, the image does not really change, i.e., you still have the outer ends higher above the middle line than the centre. How is that even possible? Well, by rotating the box, the image of an outer point of the horizon moves not only from boundary to middle of screen, but also closer to the pinhole!
Some back-of-the-envelope numbers: Standing on the ground, the horizon is about 3 arc minutes below "eye level". So in a perfectly aligned shoe-box camera of length 35 cm, the horizon will be about 0.3 mm above the center line. And if the box is 25 cm wide, the outer edges of the screen will be just about 2 cm further from the pinhole, and the curvature of the horizon line will be very subtle: only about 20 µm difference at the outer ends!
If you make a digital photo (standing at the beach, say) with a decent camara and a short(!) lens, you can digitally distort the image by making it only a few percent of its width while keeping the height. You should be able to see the curvature in the result (a curvature that was there before, but was too subtle to notice). Make yourself clear that this curvature is not caused by fish-eye effects of the lens (which do not play a role for a line though the image centre). Indeed, it happens in the same manner if you start with an upside down photo.