Esp8266 + Atmega328p as flash
It can't be done (also, why would you think that would ever be a good idea?!)

The 8 pin chip on the ESP-01 is a 512kB SPI flash that runs at (at least) 40MHz.  So to emulate that you would need:


More than 512kB of flash,
A slave SPI interface that can operate at a minimum of 40MHz, and
The knowledge, time and energy to write the firmware to emulate an SPI flash chip


The first two are certainly beyond an ATMega328P with 32kB flash and an 8MHz maximum speed SPI interface. The third is probably beyond you since you have to ask the question in the first place (which shows a certain lack of knowledge on what it is you are proposing).

There are plenty of examples for creating a web server using an ESP8266 using either the standard AT firmware or by directly programming the ESP8266's flash chip through the ESP8266's built-in UART bootloader.  Personally I find the use of AT firmware clumsy at best and prefer to always program my ESP8266's directly.  If you feel the need to also add an ATMega328P into the mix for some reason then the communication protocol that you use between them is entirely up to you.