As others have stated more eloquently, a web presence will not make or break your application, but it can definitely help convey additional information, which could be an advantage, since people generally like what they (think they) know, and it helps anyone interested to know more about you. As most of today's search committees probably don't have a majority of millennials, a prominent presence on Twitter (and likely most other primarily social media) is unlikely to help you there, so I would stick to some form of web page. (On the other hand, if you are later thinking about attracting students/postdocs, popular social media can improve the visibility of your research.)
Consider also the longer view – a website can enhance your reputation among the wider community if you put useful content on it, which could come in handy when it come to promotion or the next search. You are in a computational field, so perhaps you have developed code that is worth sharing with others. You could put that on a web site dedicated to your research, or (for more complex code) on a specialized place like GitHub, GitLab, Bitbucket, etc. (and link there from your web site). Think about where you would look for information about your research, and consider if there are ways to contribute to those places.
My hope is that my research productivity will stand on its own and I will not have to engage in social media self-promotion.
The most important thing to keep in mind is that you are not promoting yourself, but your research! Show that you are excited about it, and share what you have found out – both through your published papers and via less formal means, which also allow you to showcase research in progress, preliminary results, and ideas you are pursuing. I would consider it in fact import to stick to your research, because too much focus on your good looks or personal rants will not help you to be taken seriously.
You may already be disseminating your working papers through sites such SSRN or arXiv.org (if not, consider it, if this is done at all in your field). A website gives you the opportunity to point to all those sources from one place.
How you create the web site (prefab or DIY) is secondary to the content - something very basic is preferable to nothing at all. Keep in mind that a site full of poor spelling and grammar that has not updated in 5+ years will do the opposite of what you want, so it is preferable to have something you can set up and maintain in minutes. For a few dollars you can get your own domain name, but if something smacks of vanity (JohnDoe.com), it may earn as much ridicule as admiration. Remember, if it is not important to the people you care about (how seriously would you take someone who judges people by the layout of their website or their domain name?), don't bother.