Do you need a website to run a successful business these days? In years past, the answer would have seemed to be: almost certainly. Granted, not all businesses operate the same way, and businesses thrived for centuries before websites even existed, but if you don't have on online presence, you are missing heaps of opportunity. That much is clear.
There is a difference, however, between having an online presence and having a website. There are so many places online where businesses can attain visibility that the need for a site would seem to be diluted - not eliminated - just diluted.
First of all, let me preface this discussion by saying flat out: I am not telling anyone to give up on running their websites. I'm not going to tell you that a website is a waste of time. I'm not going to say that a Facebook page is better than having your own site or that a Twitter account can provide everything you need, eliminating the need for a site.
I will say, however, that with a growing number of tools and resources out there, it is possible to run a business without a site. Clearly, if you want maximum control over the center of your online presence, your best bet is to have your own site. That said, I don't think it's required to build customers and make money.
Want visibility from search? If you're a brick and mortar, you have things like Google Place Pages, which Google is heavily emphasizing in search results these days. Selling things online? There are a variety of e-commerce options available out there from eBay to Amazon to Etsy, and countless others. Marketing? Try building a presence across the online communities where your customers and prospects are. This could be Facebook, Twitter, forums, etc.
Where do you drive these customers? That depends on what you have decided you want the center of your online presence to be. It might be a Blogger blog. It might be a Google profile. It might be a Facebook page. It might be an eBay store. It might be something else entirely. It might be your brick and mortar store.
The point is that there are ways to be found without a site. It's all about what you market. Facebook Pages are just as marketable as websites. If you want to put an Etsy URL on a business card, you can do that, and if that business card is going to drive a customer to your business, the URL isn't necessarily going to make much of a difference.
All of that said, a website can still bring a level of credibility to a business. It's a good idea to have a yourbusiness.com or something like this for people simply looking for your business specifically. If they can't find your site when they're actively looking for it, it's not going to reflect well.
However, if you're resources are extremely limited, developing a site (or hiring someone to do so) might be more trouble and costly than it ends up being worth, particularly if you don't maintain that site - and a site is a lot of maintenance, particularly as it grows. A lot of these third-party services can reduce a great deal of the hassle that comes with running your own site.
We've been having a discussion on our own Facebook page about whether or not social media profiles/pages can replace websites for businesses. Some think so, others think not. Perhaps the question was phrased too narrowly from the beginning (leaving out a more broad set of services as discussed in this article), but I think it's still a conversation worth having. Feel free to jump in here or discuss in the comments of this article.