Today, besides certain watches etc. having collector value, the artifacts are somewhat useful as "check sources" for any geiger counters you might have lying around.
Unfortunately, Radium is not very safe to keep around unless precautions are observed. It must be stored in a well ventialted area, outside of the home or laboratory, and never in a basement. The reason for this is explained in the chart below, which I ruthlessly swiped off the internet.
Radium-226 decays eventually to Radon-222, a heavier than air radioactive gas shown as 222Rn in the chart. The gas must be ventilated out away in the environment. This is perfectly normal and natural because Radon occurs naturally.
If the gas is not ventilated but kept in concentration, besides not being healthy to breathe, it further decays into a series of Lead, Bismuth, and Polonium. When this happens, these atoms are no longer a gas, but solid elements, and they form a thin layer of dust where the atoms settle. The inside of old radium instruments and watches is covered in this. Radium's half life is thousands of years, so for realistic purposes, it does not matter how old the Radium-containing item is. The Radium is still going strong even though the phosphorescent paint it was mixed with has long ago been burned to the point where it no longer glows. Radium's own radiation killed Madame Curie. Experiment safely.
I have been told that radium, in a sealed up container, will eventually liberate enough radon to build up pressure in the container to the bursting point. This is brought up because some geniuses think to make a check source they don't have to worry about the radon with. It might take a very very long time to produce that much radon, but why do something foolish? Don't do it.
Disposal: The EPA says that you may enclose a radium-bearing clock, dial, knob, or whatever in a plastic bag and throw it in the trash. If you have a broken instrument, you might have touched the radium paint or dust. Wash the hands, and avoid ingestion of any dust or paint scrapings (no eating or smoking etc. until you are done). Plastic or rubber gloves can be used to clean up the little mess and then once it's all in a bag, put the gloves and any cleaning materials in, and seal up the bag. Then throw it in the trash and wash hands thoroughly. Check hands and area with geiger counter.