Well, if you want your website online and potentially accessible to the millions and billions of people who use the Internet, then you definitely need a web host. A web host basically provides you with the ability to place your website files on their web server which is connected online to the Internet constantly 24/7. Every time someone goes to www.yoursite.com (whatever your domain name is) the web server of the hosting company sends the user's browser a copy of your website making them able to see your website. So yes, you need a web host of some kind to have your website online, whether you pay a web hosting provider or you act as your own web host.

If you have your own web server that is always connected to the Internet and you have an extremely high bandwidth then you can potentially host your own site personally without a web host. To do this effectively, especially if you have a high traffic website is very costly. You have to pay for the computer (the machine you plan to use to host your web site), you have to possibly pay for web hosting software (for instance Windows web server vs less-expensive or free linux application server), you have to pay for extremely high speed internet with very high upload rates and then you have to manage the website making sure it is online always and troubleshoot when problems arise.

It can be thousands of dollars in all to host your site by yourself without a web hosting company, so unless you have an absolute need to do so you are most probably better off spending less than $10 a month for a cheap shared web hosting plan.