I Have a 2005 Hyundai Elantra Thats losing water, overheating and wont start when its hot. No leaks no white smoke.

SOURCE: 2003 hyundai elantra gt overheat

If the upper radiator hose is cool, but the engine is overheated than this could be a sign that the thermostat is stuck. By the way, the thermostat is at the end of the upper hose in the metal housing. There are two bolts holding it to the engine.

SOURCE: 2002 Hyundai elantra overheating.

Sounds like the waters not flowing correctly could be thermostat or other blockage.Are radiator fans working when vehicle is at operating temperature?Generaly this is a thermostat fault or faulty temperature sensor.

SOURCE: 2004 hyundai elanta i changed radiator coil cam sensor plugs wires head gaskets after overheating car still wont start

posibly starter is failing or timing may be off . . see if you are getting spark and check for fuel delivery

SOURCE: Overheating, don't know why

The most common problem with overheating is the thermostat. It becomes plugged and won't allow enough coolant to pass through the system. Another cause could be old coolant. Flushing the coolant periodically (when the coolant appears to have rust in it or has faded in color) is a good idea to keep the antifreeze/coolant up to peak levels. Bost are worth looking into but I'd be surprised (barring any leaks in the sytem) if the thermostat wasn't the culprit here. Best of luck.
Greg

SOURCE: where is termostat in hyundai elantra

first try the radiator cap..if it is bad, it will let the water run directly into the reservoir bottle then out on ground. Also take reservoir bottle out and check it for holes, if it has a hole and good cap it will let overflow drain on ground and not be pulled back into the radiator.