I do recall someone actually managing to run Linux on an Arduino. It was quite impressive, involving a very large amount of external DDR RAM and bit-banging the interface to it. Basically emulating a virtual computer within the Arduino. I can't find the original post though - it's been lost in time...
So yes, technically it is possible to run Linux, and thus Apache, on an Arduino.
Is it recommended to even think about trying to do something like this though?
The Arduino is a microcontroller, not a computer. It is designed for controlling things (hence its name) not for general computing tasks.
It can, though, run its own web server software - that is, you can connect some form of network interface (Ethernet, WiFI, etc) and implement a very very basic web server. It can't really do much, and all interactivity has to be hard coded into it in C++. There doesn't (yet) exist any simple scripting engine or anything like that (PHP, etc) and you are very limited on how much you can implement (lack of memory) as regards anything fancy.
In general a web server running on an Arduino is used as a back-end to a proper web server and interacted with using Ajax calls or other similar methods. More commonly though a web interface is shunned in favour of much lighter weight protocols or push technologies like MQTT.