I am trying to install a ceiling light fixture in a home built in 1926. The existing electrical box is very shallow (less than 1 inch) and I am having a very difficult time trying to install a new light fixture because there are not any threaded holes available to attach a swivel crossbar. What is the easiest way to install this new flush monted light fixture?

If it is the shallow "pancake" box I am familiar with, there should be a large threaded stub in the center. You can use whatever fittings, couplings, locknuts, nipples, to extend this stub down to be able to attach the fixture bar. Most likely thou, with a swivel bar, this is not possible. Also possible the fittings needed may not be readily available.

You may use long wood screws to attach the fixture to the wood framing members or use toggle bolts to attach to the finish, and not attach directly to the metal box.

If there is an attic space above it and you can get to it without much trouble

It may be a good idea to place a 2"x4" across between the joists and secure it

The electrical box was not intended to hold the weight of a fan

Then carefully place screws in the support you installed to hold up fan

Biggsie, the question is about a light fixture, there is no mention of a fan.