If you're comfortable around a command line, if you know what directory permissions are, and if you know how to configure a routher, then you can run a webserver. It's remarkably easy to do. Unfortunately that is only half of the puzzle.
Most people want to upload a website once and forget about it for a few years. If you're running your own webserver then you can't really do that.
I write this article because I have a friend whose server just got hacked. He wasn't giving the server the care it needed. Webservers need frequent attention. If you run your own server then you don't have a support phone number. You need to monitor it's performance and uptime. You need to make sure it's secure. You need to monitor the logs. You need to make sure the software is patched and up to date. You need to follow the news to find out when you should patch the server. If the server reboots will your software restart with it? If you're away from the server on vacation then who is going to do this all for you? Of course you always need a data backup strategy, but if you're running a server you REALLY need data backup strategies.
If your server is in the cloud then you don't have to worry about this paragraph (you just have to worry about if your cloud is experiencing downtime). Otherwise you need to secure the server's physical environment (Did your house flood? Is it a hot summer that's causing the server to overheat?) If the hardware fails are you going to run to the store and buy a replacement? Do you have a failover strategy in case your power or internet goes out, possibly for days on end?
If none of that scares you then there are several advantages to running your own server. You can put whatever software you want on it. You can share resources however you want. You can restrict access to the outside world. You make things MUCH more secure than standard shared hosting. You can better optimize hardware & software. You have complete control. You just need to take care of the server.
To conclude, I love running my own servers but it's not something I would recommend unless you REALLY need to.