I'm installing a hanging light fixture into an existing junction box (installed 1998) that currently does not have light fixture connected to it. In the box, there are two bare copper wires twisted together, two white wires and a copper wire twisted together, and a red and a black wire twisted together. The hanging lamp has a regular looking cord, but I've removed the "plug-in," (which was a temporary one installed in this display model), and stripped the insulation.

Which wires from the junction box should I use to connect to the chandelier? Does it matter that the wires on the chandelier are not marked with any color?

The Chandolier should have a smooth side of the wire and a half with ribs/ridges. The ridges/ribbed connects to the white, and the smooth would connect to a hot or switched leg(black, red, yellow, orange)anything but green, bare, white, or gray.
Maybe use a test socket or meter and see which goes hot with the switch.

two white wires and a copper wire twisted together,

That is just wrong.

Your wire count does not add up. If there are three bare wires there should be at least three white wires, three blacks and a red.
If there is only two bare wires there should be two white wires, two blacks and one red.

After further inspection, one of the copper wires I reported before is actually a black wire in the box. It is much shorter than the others and I could just see the end. It is not in a wire cap was painted over with white paint and was very hard to see the black insulation just barely peeking into the junction box. So, I have:

* 2 bare wires
* 2 black wires
* 1 red wire
* 2 white wires

I checked with a voltage meter and the red wire goes to a switch that already controls lights in the adjacent kitchen. One of the black wires is controlled by another switch in the same switch plate.

So I'm assuming if I connected the hot black wire and a white wire to the lamp everything should work? If I connect the red wire, it seems like the switch that controls the kitchen lights would also control this light.

I cannot follow the way you are describing this.
Check for voltage between the black and white of the two wire cable. Check for voltage between the black and whit of the 3 wire cable and between the red and white of the 3-wire cable.

Earlier you stated:

two white wires and a copper wire twisted together,

Is that statement corret? If so you have a problem.

Sounds like you have a 12-2(bare, black, and a white) and a 12-3(bare, white, black and a red.
If so, tie the 2 bares and a ground wire to the box, connect the 2 blacks together, connect the 2 whites together, and this connects with your lights white. The red(switched wire) goes to the lights blackl(likely black).

What concerns me here Strat,is that a ground wire may be being used as a neutral.

Hk, there are things still fuzzy. Black not being used, and the neutral and grounds together. I don't see what is to be gained by connecting the grounds and neutral?

Stratmando - If this helps any, if I disconnect the two white wires that are twisted together, the lights in the adjacent kitchen no longer work. I've attached a drawing.

I agree, that's why I said:
Sounds like you have a 12-2(bare, black, and a white) and a 12-3(bare, white, black and a red.
If so, tie the 2 bares and a ground wire to the box, connect the 2 blacks together, connect the 2 whites together, and this connects with your lights white. The red(switched wire) goes to the lights blackl(likely black).

Especially this sentence:
Connect the 2 whites together, and this connects with your lights white.
I believe your black is constant hot, the whites, neutrals, and the red from the switch to the light.
What wires are in the switch box?
The

Thanks for all the replies here. Here's what ended up working:

* Black wire from junction box to black wire on lamp.
* White wire from lamp to two white wires from box.
* Keep red and black wires (both from box) connected.
* Keep both cooper wires from box connected.
* Attach ground from lamp to green screw on bracket.