Older motherboards with electrolytic capacitors can fail when one or more of those capacitors dry up, leak, bulge, or for some other reason drift outside allowed tolerances. It’s possible to replace any obviously failed electrolytic capacitors, but even if you do this, others may be on the verge of failure. Replacing the whole lot is uneconomic. A better solution is to replace the motherboard, or even buy a whole new system.
A new motherboard will probably have solid-state capacitors, be much faster, accommodate more memory, provide USB 3.0 ports, DVI-D, HDMI and / or DisplayPort video connections, etc. However, after taking into account the cost of a new motherboard and the labour required to remove the old one and fit the new one, a whole new system might make more sense. This will definitely be the case if you have trouble sourcing a new motherboard that will accommodate the existing central processor.