https://en.wikipedia.org/wiki/Timing_attack
In high level languages like Rust, the compiler does not prioritise trying to emit machine code which executes in constant time for all inputs. OpenSSL has implementations for some primitives which are known to be constant time, which can be important.
One option if you're working with Rust anyway would be use something like Ring:
https://github.com/briansmith/ring
Ring's primitives are just taken from BoringSSL which is Google's fork of OpenSSL, they're a mix of C and assembly language, it's possible (though fraught) to write some constant time algorithms in C if you know which compiler will be used, and of course it's possible (if you read the performance manuals carefully) to write constant time assembly in many cases.
In the C / assembly language code of course you do not have any safety benefits.
It can certainly make sense to do this very tricky primitive stuff in dangerous C or assembly, but then write all the higher level stuff in Rust, and that's the sort of thing Ring is intended for. BoringSSL for example includes code to do X.509 parsing and signature validation in C, but those things aren't sensitive, a timing attack on my X.509 parsing tells you nothing of value, and it's complicated to do correctly so Rust could make sense.