No, it's not a problem at all. There are already several high-performance databases written in Rust.

Lifetimes are mainly a learning barrier for new users, and affect which internal API designs are more convenient, but they're not a constraint on the types of applications you can write.

Rust strongly guides users towards using immutability and safeguards uncontrolled shared mutability, but you can use shared mutable memory if you want. In single-threaded programs that's trivial. In multi-threaded programs shared mutability is inherently difficult for reasons beyond Rust, and Rust's safeguards actually make the problem much more tractable.