New here. So for a project I am looking at using an ESP8266 as a server and hosting live pressure sensor data.
I wanted to know how feasible this is, as well as what direction I should start moving in as I cannot find anything similar to this so far.
Would really appreciate any guidance 
Here's a way to do that. This is for the 32, but a similar approach can be used with the 8266.
"This is useful to send updated sensor readings to the browser, for example. Whenever a new reading is available, the ESP32 sends it to the client and the web page can be updated automatically without the need to make additional requests."
ESP8266 DS18B20 Sensor Web Server Arduino IDE (Single, Multiple) | Random Nerd Tutorials
You find several esp8266 sensor web servers on this site.
Welcome to the forum. You might begin by reading this tutorial.
It does not do precisely what you wish, but if you replace the digital inputs with analog inputs (assuming your pressure sensors are analog), this should be a good start.