Is it 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