EndGame: you'll get over it.
There's a thread somewhere where I describe four kinds of relationships that people can have with mathematics: practical, constructive, creative, and anxious. If I were to describe similar relationships people can have to science, you would fall into the anxious category.
Basically, while your epistemic quandry is well-founded, it's not really worth your time.
For one thing, why are you willing to trust what your eyes tell you, but not what a history book tells you? Turns out your brain translates the world into experience in a fairly obtuse fashion. Not everything you see is real, and not everything that is real is visibile to you.
What evidence do you have that under the hood of your car is an engine? Just memories; memories can be false, and often are. What evidence do you have that there's no purple dragon hovering over your shoulder? If you turn to look, it will disappear. What evidence do you have that the wall in the background is the color it appears to be? You're not really seeing the wall; your brain is making an assumption about what it looks like.
For another thing, there are plenty of aspects of the world that you have no direct access to. Do you believe in subatomic particles, or distant galaxies? Like you said, you don't have to; it won't affect your life. That being said, you might as well believe in them, at least in the interim, since they give you a slightly greater understanding (even if potentially false) of what's going on around you.
To make a long story short, there is no proof or truth in the real world, but that doesn't mean that everything you believe is entirely on faith. There is a degree to which any new claim is consistent with your existing beliefs and with your observations of the world. You can use reasonable, if not always rational, methods to sort through claims and find the ones that you want to accept. In the end, of course, you do have to make some "irrational" assumptions:
1) Induction is a good way to get knowledge, because it has usually worked in the past.
2) Seeing is believing, but what somebody else saw is pretty good too.
3) Theories are good to the extent that they make testable predictions, but barring that, judge them on parsimony.
Hmm... those are the only ones that come to mind at present, but it seems that you can get a lot of science done if you're willing to accept those assumptions.
My last point is that, as interesting as these are to think about, you shouldn't let them affect your life; apples will still fall to the ground whether you believe in gravity or not.
-Erasmus