According to everything I have found on the web and experienced with my own watch, the watch will not register as mass storage so that you could access the recorded logs directly. Therefore you cannot just plug the watch to a random computer and copy he logs like it was a USB thumb rive.
What you can do is use Moveslink offline, i.e. you don't have to connect it to the Movescount online service at all or if you have temporarily no internet connection. In this case Moveslink will save the logs locally until it is connected to Movescount in a proprietary
xml encoded file. This file by itself won't be of great use in the first place but some nice people already have written a tool to convert the logs to a
gpx file. I haven't tried it myself but if I read the documented options right you can choose to just export the GPS positions or other (heart rate etc.) data. According to https://code.google.com/p/ambit2gpx/wiki/Readme the logs are stored in the following location:
When transfering data with moveslink, an xml file is stored on computer. On Windows, it can be found on locations such as:
C:\Documents and Settings\'username'\Application Data\Suunto\Moveslink2 (on Windows XP)
C:\Documents and Settings\'username'\AppData\Roaming\Suunto\Moveslink2 (on Windows 7)
Pay attention to the fact these 'Application Data'/'AppData' folders are hidden by default as they are 'system folders'. Change settings of Windows explorer if you want to make them visible.
So to conclude the the information above: You can use Moveslink offline to download the logs from the watch and there's a tool to convert the files to
gpx format. But you will need a computer where you can install both tools therefore you won't be able to use any random computer unless you are able to install the needed software there.