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Radiation comes in different forms and its ability to affect electronic devices depends on its ability to penetrate the electronic equipment and then to penetrate the packages with semiconductor devices in them. Usually it will be beta and gamma radiation that will have this ability; alpha particles will usually be stopped by outer packaging very easily. The common quality that is measured in Radiation is its ability to ionise materials. In semiconductors this ionising radiation can have two major effects: one is to produce electron-hole pairs which can create "soft" errors (errors in operation but not permanent damage) and, if the radiation is sufficient, permanent damage by creating large numbers of charges with sufficient energy to be injected into Silicon dioxide regions (where they stick) and change a transistors characteristics. Such high levels of radiation can also disrupt the crystal lattice and damage the transistors in that way. Normal semiconductor devices such as those in a typical computer would have sufficient soft errors at relatively low levels of radiation to render the computer unusable though not necessarily cause permanent damage. These levels are not generally sufficiently low that you would want to stand around in for long in it either! I suspect this is the sort of levels that we may be getting close to at the site in Japan at present. At the next level up where humans would be seriously harmed is similar to the point where normal semiconductors also get permanent damaged. However, it is possible to make semiconductor devices that are very resilient to radiation - at least for a period of time. This involves different processing and careful design and, as a result, they are not cheap to make. Typically they will use a silicon-on-insulator process and complete computers can be made (and are made for military applications) that can withstand around 1 megarad, which would be lethal to a human. I don't know what levels were reached at Chernobyl but I would guess such semiconductors would have worked for some time there.
The problem would be that it would take some months (or years) to design and build a suitable "robot" from such a set of components (which may or may not be readiliy available).
Not to contradict any of Graham's points, but realize that radiation is all around us, it always has been, and just about everything is a source of radiation of one sort or another.
Geezer is right. Cosmic ray strikes do disrupt electronic devices on occasions and if a device is intended for high altitude use, and space operation in particular, it has to be made radiation hard to some extent and/or fault tolerant. This is usually to be able to tolerate "soft errors" I mentioned earlier. Sufficient radiation, that damages devices, cannot be countered by such means and a radiation hard device is required (and this may be required for long term satellite operation for example). The very rad-hard devices are primarily for defence usage.
Yes, the technology exists, but it takes time and money to invest in putting it altogether for a specific application. I guess nobody was planning on these nuclear incidents occurring.