Don’t get me wrong, there are plenty of successful people with pretty generic email accounts. I worked for few, some have been clients. However, these are companies or people who have established themselves with reputation and rely on the Rolodex as major source of business. Similarly, there are plenty of people who buy up email accounts to perpetuate the appearance of legitimacy in their scams. There will always be exceptions, but as a general rule, if you are starting out your company, I highly recommend that you spend the few pennies a month and buy yourself an email account with a domain related to your business and your webpage. Think of it as brand recognition or think of it as keeping business separate from pleasure. No it will not attract customers, but at the very least it will help retain attention and confidence in those seeking out your services.
All this said, it is purely my opinion, so please feel free to correct me. As a matter of disclosure, I actually have 4 email accounts. A free hosted (MSN) personal account, a free hosted work account (Gmail) that operates as my business catchall email client (my web based mail client that syncs nicely with my Android phone), my law firm email address, an email for the company that I work with (new companies require more email addresses sadly), and lastly a free hosted spam account to dump those occasional newsletters, IQ tests, and other free goodies. It may sound like a lot, but once you have it organized, it operates pretty automatically.