is there possible to install curl into busybox in kubernetes pod

The short answer, is you cannot.

Why?

Because busybox does not have package manager like: yum, apk, or apt-get ..

Acutally you have two solutions:

1. Either use a modified busybox

You can use other busybox images like progrium/busybox which provides opkg-install as a package manager.

image: progrium/busybox

Then:

kubectl exec -it busybox -- opkg-install curl

2. Or if your concern to use a minimal image, you can use alpine

image: alpine:3.12

then:

kubectl exec -it alpine -- apk --update add curl

No. Consider alpine as a base image instead that includes BusyBox plus a package manager, or building (or finding) a custom image that has the tools you need pre-installed.

BusyBox is built as a single binary that contains implementations of many common Linux tools. The BusyBox documentation includes a listing of the included commands. You cannot "install" more commands into it without writing C code.

BusyBox does contain an implementation of wget, which might work for your purposes (wget -O- http://other-service).