What should I do? I work at a tech company who sucks at tech.
pedalpete: I think it depends on how you feel about the company. Why did you take the job originally? was it because the company was a global multi-billion dollar brand? or was it because you thought you could make a difference.Clearly there is opportunity for improvement. If you are capable of making those improvements, and leading a change, this could be hugely beneficial to you in the future. Either with the current company, or when moving to a new company. 
Taking initiative (from my experience) has regularly paid back in spades, and where it hasn't it has been an amazing learning experience.Taking on each of the issues you mention
1) This is not entirely a surprise. Excel is surprisingly easy to use for scheduling, and you'd be surprised how many people don't know that there are alternatives. Maybe something as simple as setting up a group calendar and teaching people how to use it would help. (I'm currently building a calendaring app which targets this whole excel thing specifically, so I've got some experience with why people use excel).2) your web sites share storage with your e-mail server? is this a SAN? This just sounds REALLY strange, and I agree it makes it seem like your CTO (or whoever) doesn't have a grasp on things. I'd attack this one directly (though politely) and figure out why this is, maybe explain the dangers, and figure out a migration plan to move e-mails to an appropriate location.3)Why is the database so slow? as you say it is a client database, so I can't imagine it is slow due to the number of records. Is the client database important? By the sounds of things, it isn't. Not everything needs to always be updated.4) I believe this issue is very common at many companies. I think it comes down to communication tools, and protocols for releasing code.As I look at what you're up against, I think you may benefit from writing a one pager about how and why you think fixing these issues is important. 
Also, I'd try to get macro, and look at not only how to solve each issue like a task list, but try to find what the common thread is as to why the company has these issues, and what framework/culture/other would not only resolve these issues and others but also prime the company for continued growth.If you can get some people within your company to listen to your plan and get some momentum, you may find yourself helping the company. If you think people in your department aren't interested, you might consider going to HR. It is there job to retain top talent (like yourself), and if you'r e thinking of leaving the company because of these issues, it is possible you aren't the only one. HR may be able to help.