You can wrap up the build and the copy with a multi-stage docker build. Something like the following.

    FROM rust:latest

    RUN rustup target install x86_64-unknown-linux-musl
    COPY . /src
    WORKDIR /src
    RUN cargo build --release --target=x86_64-unknown-linux-musl

    FROM scratch

    COPY --from=0 /src/target/x86_64-unknown-linux-musl/release/rust_docker_barebones /rust_docker_barebones
    ENTRYPOINT ["/rust_docker_barebones"]

Although, I typically recommend _not_ using the scratch container unless you really need something that tiny. It makes debugging very difficult if you don't at a minimum have a shell and a package manager.

Also, packaging up rust to be static like this can take some extra steps. For example, loading in root certificates.