Does water only run in veins under the ground? Or is there a depth where there is water everywhere?
Water is present in specific geological formations. There is no specific depth where water is present everywhere.
The idea that water runs in "veins" is an old concept that is actually rather wrong.
Most of the underground water is retained in formations which porosity allows for water to percolate inside. As example sandstone, gravel or fractured limestone. These formations act as a "sponge".
An underlying layer of impermeable rock (such as clay) acts as barrier allowing the water to be accumulated within the hosting formation (reservoir).
In some cases the water is pressurized and when a well is drilled it will spontaneously flow. It is called an artesian aquifer.
In some other settings water can actually flow in underground rivers, As example the complex caves systems below Florida (USA) host large amount of groundwater.
Generally the deeper underground the more likely is to find a saturated zone filled with water.
In conclusion water is quite widespread underground but we cannot state that there is a specific depth after which water is present everywhere. You can read more about groundwater in the Earth Science topic here.