I will be "stuck" in Doha (Qatar Airlines) for about 8 hours during the night. Is it possible to leave the airport and visit Qatar for a few hours with a standard (Belgium) international passport?
Yes, as listed on wikitravel, Belgium is one of the countries that gets a visa on arrival. The costs are around 22€.
Qatar issues a visa on arrival at Doha's airport to passengers who are citizens of Andorra, Australia, Austria, Belgium, Brunei, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Iceland, Ireland, Italy, Japan, Liechtenstein, Luxemburg, Malaysia, Monaco, Netherlands, New Zealand, Norway, Poland, Portugal, San Marino, Singapore, South Korea, Spain, Sweden, Switzerland, UK, USA and Vatican City. The price is QR 105, payment by credit card is accepted, and grants a 30 day stay.