For what you describe running your own server, with all the associated costs and headaches, wouldn't be at all worth it. Pretty much any low- to mid-range shared hosting service should get you off to a great start, and they only cost between $10 and $50 per month, depending on the hosting package and provider.
There are several common web applications that seem to meet your criteria. The first one would be a web forum
very much like Bleeping Computer here. As you can see, users (like yourself) are able to enter textual data of almost any type and it will be stored in a database indefinitely, wherefrom searches of the stored texts can be made. There are a number of free and paid-for forum software packages available, and many shared hosting providers offer one-click installs.
A related web application would be the guestbook
. Guestbooks have fallen out of favor somewhat in the last 15 years, but there are a number of free guestbook widgets still floating around. From Wikipedia: "A guestbook (also called guest book or visitor log) is a paper or electronic means for a visitor to acknowledge their visitation to a site, physical or web-based, and leave their name, postal or electronic address (if desired), and a comment or note, if desired."
A third option would be to write a simple custom web app. This would give you the greatest control over how information is collected, searched, and displayed. This does carry with it the requirement that you, or someone you employ, knows how to write a web application. It's not terribly difficult, but it would certainly slow you down to have to learn from scratch.