I was wondering whether smart contracts can handle two function calls at the same time or one of them has to execute first before the second.
Solidity is a deterministic language, meaning it only runs one function at a time and at a deterministic order.
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Okay. I got it, thanks Koray 
I disagree with @Koray_Kaya. This is not so much about Solidity as a language but about the EVM as a runtime. Indeed, the EVM does not allow parallel execution, but this can be deceiving. Concurrent execution is still possible through reentrancy, and it's no less deterministic.
If contact A is executing foo and it makes a call to contract B, B can now call A again and execute function bar. In a sense, at that point, both foo and bar are executing concurrently.
This is actually a very important thing to understand, because it's how reentrancy attacks become possible.
Isn't that the same as saying if a function call is made inside of a function, both functions are running concurrently? Indeed two functions are being ran, but in no way are they running at the same time. Never would for example a race condition occur.
E.g. foo in contract A executes bar. How is this different from your example?
Race conditions can't occur.