The short answer: It's darker then you think it is. Here's a depiction of various brightnesses and an an exposure value which nominally will give correct exposure at that brightness. Note that these are overlaid — the area of the whole circle is what matters, not the separated rings.

This seems shocking, because our eyes are so good at adjusting, but sunlight is 500× brighter than a typical home interior.

Your settings of "f/5.0, shutter 1/160th, ISO 6400" correspond to a correctly-exposed image at EV 6 — exactly what's labeled as "home interior" on this chart — so theory is meeting reality.

EV 0 corresponds to a one second exposure at a theoretical f/1 and ISO 100. You can calculate in stops from there by hand, or use a site with an EV calculator.

Note that higher ISO isn't, itself, what's causing the noise. That's really caused by lack of light, and, short of getting more light in, it's actually not a bad thing. See this answer for more detail and practical examples.

If you want a lower ISO, you'll need either slower shutter speed, wider aperture, or more light. If you want to go down to ISO 800, you'll need three stops to compensate (because you halve 6400 three times to get to 800). That means you could get equivalent exposure from f/5.0, ISO 800, and 1/20th of a second. You'll probably need a tripod, or good image stabilization, to get acceptable results, though — and if you're shooting people, they'll have to be very still. Opening your aperture all the way to f/4.0 gives you another ⅔ stop — but that only gets you to f/4.0, ISO 800, and 1/30th.

My suggestion is to invest in a low-cost off-camera flash, either optically triggered or as a radio slave.