CRT Repair Cautions
DANGER CRTs like many mains powered devices contain hazards that have the potential to be lethal. Do not attempt to repair a CRT (or other mains powered device) unless you have a good understanding of the hazards.
CRT Operation
CRT stands for "Cathode Ray Tube", where the cathode generates an electron beam that is moved across the phosphor coated tube to produce light. This is somewhat unintuitive with the concept of "conventional current", the idea that current flows from a positive charge to a negative one which is the way most electronics is taught and presented. The reality is that electrons actually flow from a negative charge to a positive one. This fact has been used for multiple types of display technologies like VFDs (Vacuum Florescent Displays) and Nixie tubes which don't use Black-body radiation to create light unlike an incandescent bulb. The electrons hit a phosphor layer in VFDs and flow through a gas in the case of Nixies that they interact with causing the glow. But VFDs and Nixies operate at dozens or a few hundred volts instead of multiple thousands like CRTs. CRTs need the electron flow to be fast and precise, so they use a vacuum like VFDs but increase the voltage difference significantly.
CRTs also have two conductive coatings applied to both the inside and outside. The inside carries the anode voltage and the outside is connected to ground. This was done to use the CRT itself as a capacitor to smooth power fluctuations in the anode voltage. The glass of the tube acts as the dielectric barrier to form the separation. The CRTs capacitance will depend on its size which increases the amount of parallel surface area.
CRT Anode Shock Hazard
Note that the below section discusses the shock hazard from the high voltage present on the anode of most common electromagnetically focused and deflected CRT displays, some specialized CRTs like what is used in most oscilloscopes or older monochrome TV sets are electrostatically focused and deflected and will have high voltages other the the anode.
As explained above the CRT anodes act as a capacitor storing energy, in combination with the fly-back output capacitance, 1 to 10 nF is reasonable to expect. This means at 20,000 V the stored energy would range between 0.2 to 2 J. For perspective an ATX computer power supply may have a 450 V 270 µF capacitor which can store about 27 J and a defibrillator may deliver 10 to 350 J to a patient. Although 0.2 to 2 J may be less than other devices a shock from a CRT is still hazardous, especially when you're not expecting it. The very high anode voltage is often greater than the dielectric breakdown voltage of things like plastic handled screwdrivers, pieces of wood or small air gaps meaning they become conductors and can shock you.
Note that you will only be shocked when you place yourself in between the anode and cathode of the CRT, as the outside conductor of the tube is typically connected to circuit ground, anywhere on the chassis of the device acts as the cathode of the CRT, touching the anode and anything connected to circuit ground will result in a shock. The only way to access the anode is under the anode cap though, the inner coating is not externally exposed. The anode cap is connected to the flyback transformer, but due to the fully encapsulated nature of most modern fly-back transformers you cannot come in contact with the anode through its connections to the PCB. So if you do not need to remove the anode cap, you are unlikely to come in contact with both sides of the CRT and be shocked. The only time it is likely to happen is when you need to remove the anode cap, which should only be needed when removing the PCB the flyback transformer is mounted to.
When working on a CRT attention should be paid to the charged state of the display tube. The dangers of being shocked are frequently characterized as being lethal, which is a gross exaggeration. You can be shocked (and the author writing this has) by a CRT and suffer no ill effects other than surprise and embarrassment. The severity will depend on the characteristics of the tube though.
Discharging a CRT Anode
Since the CRT itself functions as a capacitor, the way to discharge it is to connect the anode and cathode to allow for the voltage potential to equalize between them. The CRT does not need to be connected to earth ground to achieve this. To do this you need to connect the clips of the anode cap while it is still attached to the tube to ground on the chassis. The most difficult part of the process is is getting a conductor underneath the anode cap on the CRT which may be "stuck" if it was applied with a grease. Using a non-conductive tool to first lift the silicone portion of the anode cap can be helpful. A simple way of then connecting the anode to ground is to use a wire with clips to connect the chassis to a flat bladed screw driver and then tap the clips of the anode cap with the screwdriver. You will most likely hear and potentially see a spark when the discharge happens.
Some CRTs may have a "bleeder" resistor installed which connects the CRTs anode and cathode. This will be an extremely high value resistor that won't allow much current to pass through, but once a CRT has been turned off it will slowly dissipate the charge. If you attempt to manually discharge and do not hear or see a spark this may be why.
Exposed Metalwork Shock Hazard
Some CRTs contain metalwork such as the frame or heatsinks that are live with mains voltage (100 to 240 V), touching those will result in a potentially lethal electric shock. When working on those types of CRTs an isolation transformer is essential but does not eliminate the risk of electric shock.
CRT Ionizing Radiation Hazards
In summary this hazard is only present in very specific older CRTs and some CRTs not intended for use in a home and even then the risk is minimal to repair.
Accelerated Electrons Radiation
CRTs have the capability to generate small amounts of low energy ionizing radiation (x-rays) when turned on due to the accelerated electrons rapidly decelerating when hitting the front grilles, phosphors and glass, however this is typically an insignificant amount as modern CRTs contain leaded glass that block this radiation, some CRTs from the 1960s emit ionizing radiation "in excess of desirable levels" so should not be viewed from close distances or for long periods. Regardless, when switched off all ionizing radiation production is stopped, no 'residual' radiation remains, there is no risk of this type of ionizing radiation exposure when repairing CRTs.
Thoriated Tungsten Cathodes
Some CRTs contain cathodes with thoriated tungsten cathodes, as thorium is a radioactive isotope it emits ionizing radiation (gamma rays) even without power. However due to the very small amount of thorium, it's containment in the glass tube and it's limited use in modern CRTs it poses no risk to repair.