The answer to your first question is a simple "yes".
With the caveat that your new Jira server must be the same version of Jira, or *higher* than your old server.
For "how", there's actually quite a number of ways to do this. The best one depends on exactly what your setup is. What is the old server basically? Yours, hosted, OnDemand, something else? Etc. A bit more detail on what the old server is running, and what you're going to, would get a better answer.
You need to contact Atlassian support to get an export of your OnDemand data. Once you have that, you'll be able to import it into your new server (Assuming the version of Jira you've got is the same, or higher, than OnDemand), copy the attachments, and it should work fine.
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