score:13
You're confused: requiring the expiration date and the 3- or 4-digit code at the back has nothing to do with being a “US credit card”.
Girocard is a German payment network. It can only be used to pay for things in Germany. You can't use it in other countries, European or otherwise.
Many Girocard cards are simultaneously Maestro or V PAY cards. Those are international networks, so if your card has a Maestro or V PAY logo, it can be used in most other European countries, including Austria I believe. Maestro is also accepted in many non-European countries, including places where only the magnetic stripe is used, whereas V PAY is more restricted and requires a chip-reading terminal. If your card has one of these logos, you can use it online, and you will probably need to provide the expiration date and the 3- or 4-digit code at the back for authentication.
Whether the card has a chip or not only matters when you're paying by inserting the card in a reader (“customer-present transactions”, in industry jargon). For online payments, authenticating you as the card holder by asking you for the expiration date and 3- or 4-digit code at the back is standard. (It isn't required, but if the merchant doesn't ask for this, they get less insurance in case of fraud, so most do.) Merchants who ask for this information are supposed to adhere to security standards to limit the security risks, including not storing the 3-digit code in any permanent form (i.e. it must directly go to the bank for authorization and be transferred afterwards), though it's difficult as a customer to know whether the merchant is compliant.
If you don't want to trust the merchant with your payment information, see if your bank issues single-use credit card numbers. Some do it as part of their standard package, others for a fee, others not at all.
If you want to pay for something on the Internet without using a credit card at all, your opportunities are limited. Bank transfers are a hassle for both the payer and the payee and have hefty international fees with little guarantee for either side, which is why they are rarely used. There are payment intermediaries (Paypal being a well-known one), which are popular for micro-payments but not so much for hostel or hotel reservations.
You can look for hostels that are prepared to reserve a room without having your credit card details. They exist, but they're rare.
Upvote:6
Gilles' answer is excellent, so I won't try to cover the same points a second time. I'd like to focus on one offhand remark in your question, though:
"I think that bank transfer would be problematic, because international money transfers take a lot of time and cost much [...]"
In some places, that might indeed be the case, although it's getting less and less common. But since Germany and Austria are both within the Single Euro Payments Area (SEPA) and use the Euro as their national currency, bank transfers between them should not be any slower or more expensive than transfers within each country (i.e. in practice, they should be free and close to instantaneous).
Of course, the booking site you're using might not offer such an option, but you could always contact the hostel directly and ask for their IBAN code in order to make the deposit. I see no reason why they should refuse to do so.