It should be noted that the term "radiation" is often misunderstood by the layman.
Radiation consists of either electro-magnetic radiation (radio waves) or energetic atomic particles.
Energetic atomic particles only come from high-energy reactions -- a nuclear reactor, the Sun and stars, and occasionally energetic lightning strikes in the atmosphere. No common electrical equipment emits these (though some smoke detectors contain a tiny bit of "radioactive" material).
Electro-magnetic radiation (EMR) comes in a spectrum, with infrared radiation (what you feel standing near a fire) near the low end, and gamma rays on the high end. Between infrared and gamma rays are visible light, ultraviolet, and xrays.
Below infrared in the spectrum is radio waves -- AM, FM, TV, cellphones, microwaves. At the very low end of radio waves are Ultra-Low Frequency (ULF) waves. Ordinary electrical power is at 50-60 Hz (compared to 500,000 to 1,500,000 Hz for AM radio) and would be classified as ULF.
Radiation can be hazardous for basically one of two reasons:
1) "Ionizing" radiation has the ability to "knock apart" molecules in the body, potentially causing cancer or genetic mutations. "Ionizing" radiation is basically either radiation from "energetic particles" or radiation in the electro-magnetic spectrum beginning with ultraviolet (which is why sunlight can cause skin cancer).
2) Any intense EMR has the ability to damage living cells via heating (hence microwave ovens). Even though the heating may be very slight and not perceptible, damage to the delicate inner workings of the cell can occur. But while the effect of "ionizing" radiation is cumulative -- low intensity for a long time is as bad as high intensity for a short time -- this is not true for EMR. Below a certain intensity EMR simply does not cause cell damage.
(The "radiation" emitted by a UPS would be EMR, of the ULF variety -- the same radiation emitted by every other plugged-in electrical device in the building.)
There is some disagreement with regard to what the threshold should be for considering EMR dangerous, but there's been at least 50 years of research in the area and no one has come up with any real evidence that the levels we experience in a common home or office environment are anyway near the harmful threshold.