How much current can a 74HC00 supply?
Unfortunately, this datasheet does not have a plot of the decrease in output voltage with increasing output current.  However, section 8.3 restates the 4 mA figure, so I would take that as a recommended upper limit for continuous operation.

Note that at 4 mA, the output voltage will not be 5 V when sourcing or 0 V when sinking.  This is covered in section 6.5