While in Belarus, you have several options to get Internet connection.
Dial-up from your computer
If you bring your computer (say, laptop) with you, it is possible to set up a dial-up connection to the Internet. You do not necessarily need a contract to use a dial-up service in Belarus. You have two options.
Many providers sell so-called "Internet cards" that have account IDs and passwords valid only for some specified period. The underside of the cards has the information on how to set-up your computer (only in Russian) to connect to the Internet.
Beltelecom "Subscription-Free" access
Belarusian state telecom company provides the service where users can connect to the Internet simply by dialing from the computer a special line and using any ID or password. Usage costs are then billed to the phone line owner. To connect dial from your computer number '8w600100' ('w' is for 'wait for dial tone' and is understood by a computer) with any username and password.
There is a network of cybercafes in Belarus that provide its visitors with access to the Internet. There are both state-run (Beltelecom) and private cybercafes that offer access to the Net as well as some related services such as printing, CD-copying, scanning, etc.