The usual sites are for longer term rents in Austria - I need a room for a month. Obviusly, a hotel would be too expensive.
What site or social media group could help me finding a room for a month, possibly in a WG?
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If you want to stay for a month in Vienna a vacation rental/Airbnb would be the most common choice. Search for "Ferienwohnung Wien" and you will find multiple websites offering accommodations (of which Airbnb usually isn't the cheapest)
Nevertheless I think your chances are slim for a WG as most landlords want long-term tenants. You best chance would be to find somebody who sub-rents their room for some time.
Using social media groups for finding housing isn't a good option if you need to ask for those groups as you're obvious not part of those social groups.
Good luck searching!