They have a shallow metal box made for fans, it would mount to the beam.

Can you mount a box to the beam? It'd have to be a box rated for fans, and you'd have to assess the condition of the "beam", which may be a piece of scrap lumber that somebody rigged up between rafters. The box might also stick out a bit. Seldom do things work out in your favor in old houses. With a little luck, the decorative shroud with the new fan will cover everything up.
Also, what type of wire is it? Are the 2 conductors enclosed in an outer sheath, or are they just separate wires (AKA "knob & tube")? What is the condition of the insulation? Is it brittle or intact?
Without attic access, you'll need your lucky stars to line up...

The two wires are covered in cloth, which is intact. The wood could be a piece of scrap lumber. However, if that's the case it is in there very firmly. Is there an easy way to determine whether I am looking at a rafter? Thanks so much for your help!

Good insulation is good news. My point about the "beam" is to make sure it's in good condition and not cracked where you'll attach the electrical box. I had a similar installation in my kitchen, where the previous owner put up a 1x6 across the rafters and hung a fan from it. Apparently, they didn't drill any pilot holes before attaching the box with wood screws. I found that the 1x6 was cracked around 2 of the 4 screws, so only 2 screws were holding up this ~15 pound fan and light assembly!
As for identifying the 2 wires, you can use an extension cord and a neon voltage tester. Plug the extension cord into a nearby receptacle. Separate the 2 wires at the fan. Turn on the power. Insert one test lead in the neutral slot (the wider slot) on the extension cord, and touch the other lead to each of the wires. The one that lights up the tester is the ungrounded (hot) and the other is the neutral (white.) Shut off the power when done. NOTE: This test assumes that the extension cord is plugged into a receptacle that is wired correctly. You can verify that with an inexpensive plug-in tester, available at any hardware store.

Can the hot wire be identified with a voltage tester? Move the wires apart, turn on the switch and test for the wire that is hot.

Thanks so much for your advice. As suggested, I got the shallow metal box designed for fan. It was exactly what I needed. Since I wanted to make sure it was attached to a rafter, and not just a piece of scrap wood, I enlarged the opening in the ceiling a bit (necessary anyway to install the box) to see what I was working with. This also gave me a better look at the wiring, and I noticed that one seemed darker than the other. So I just attached the fan based on that. Everything is working great! This was my first ceiling fan installation. Thanks again for your help!

The hot wire really should be identified, you want the switch to cut of power going from the switch to the fan not coming out of the fan.