We have always welcomed questions like this (on internet related topics!).  It's better to make enquiries than to make an uninformed decision.  We don't want to see you fall for a scam.  Nine times out of ten the answer is: make your own decision but our opinion is "don't do it", usually backed by experience/research (but perhaps I'm just a cynic).
In fact the reason we've been asked is often because you already have a doubt but haven't spotted the flaws.

Your email password may be as valuable to a hacker as your bank account details.

There are several ways they can expliot it - at the simplest, your saved messages can tell someone an awful lot about you, including passwords to other sites.  That's a great starting point for them to impersonate you on-line.

Having a problem accessing a website? The most common problem is a fault with the web-server.  Even the most high-profile businesses have problems sometimes.  A couple of weeks before I wrote this HSBC had a major problem for a few hours.  However other problems may be with internet routing or local to your PC.  The MORE link explains more and suggests some actions you can take.

If you have a website you will get calls and emails promising to "get you to the top on Google".  The SEO business is swamped with tricksters, don't be taken in.