The answers given each seem to say “here is what you need to do: bla bla bla…” Lots of trees; might be hard to see the forest. You might find the following useful:
There are basically three technologies you might leverage to do what you want to do, which is allow authorized users to access Windows files which sit on “regular office computers” (not file servers) securely from “anywhere on the Internet.” Within each technology there are many many products, ranging from free to expensive, from already-installed-with-Windows to things requiring full time IT staff just to turn them on (did I hear someone say “SharePoint”?). Here is a list of the different technologies:
Native Windows file sharing (called CIFS) over a VPN. Lots of information above. The key thing is that if your users will ever need to access files from unknown network environments (think: hotel Wi-Fi), you will need to hire a bodyguard to keep them from killing you when they return from their trip.
WebDAV. This requires no VPN, and it uses HTTPS for transport, so it works as well as web-surfing does (that’s a LOT better than SMB, promise). But of course you need to configure the firewall to allow the office machine-with-files to be a web-server.
SFTP (secure FTP) or even FTP over a VPN. Internet Explorer is an OK FTP client, and (s)ftp works over the Internet (!) -- but you don't exactly "access" the files, you transfer the files. If that's what your people need to do anyway, this might be the cheap-and-cheerful choice.
However, I believe that Dropbox is WebDAV-as-a-service. So if your users are used to and want the “web-folder” UX metaphor, they might find Windows file sharing (CIFS/SMB) too hard to do, SFTP too bare-bones, and WebDAV “just right.”
You can find lots of products by searching for CIFS/SMB, WebDAV, or SFTP.
In the Linux world there is an open-source implementation of SMB/CIFS called Samba. But you don’t care about Linux, I think.