There is very good evidence that static electricity is damaging to modern microelectronics.
Some components are more sensitive than others - CPUs and graphics cards are particularly sensitive because their chips are built on the latest very small processes. Hard drives are very sensitive, because of their delicate heads. These sensitive components may be affected by very low static charges (less than 500 V) which are unlikely to produce any sensation at all.
Motherboards are generally less sensitive (chipsets and IO controllers tend to be built on older lower precision machines), but nevertheless need some protection.
The problem with damage from static electricity is that it is often very subtle and leaves no externally visible signs. In many cases, it can cause a very slight degradation which is not immediately apparent - but which can knock several years off the life of a chip, or which might cause instability at high temperatures.
If you are doing this as a business, especially if you are repairing computers, then you do need to take static precautions. If you repaired a customer's machine, and 3 months later it broke down - if the customer knew that you didn't take anti-static precautions, they could very easily blame you for damaging the computer and demand that you repair it. Unless there was a visible fault with the machine (e.g. something mechanically broken or connected wrong, it will be essentially impossible for you to prove him wrong). Similarly, if you are building machines, failure to take adequate anti-static precautions could very easily end up with a number of your machines breaking down prematurely - leaving you with a poor reputation, and lots of warranty claims.
You don't need to go to extreme lengths - e.g. antistatic flooring, antistatic clothes and shoes for staff, antistatic work surfaces, and grounded wrist bands. However, at least, you should have grounded wrist bands and antistatic work surfaces (or antistatic mats).