I just want a linux distro that boots fast, has a bash terminal (no gui) and busybox.
Is there any such distro or is it hard to compile a kernel and just make one myself?
What you are searching for is called ttylinux.
The distribution was hosted at Ttylinux.net, and is currently not maintained.
You could always try DSL (Damn Small Linux)
I don't think any of these explicitly come with busybox (I don't know of any distro that ships with it), but it shouldn't be too hard to install it afterwards. From what I understand your main concern is a small fast system with a decent shell.
The best most current distribution I can think of is Debian NetInst: http://www.debian.org/CD/netinst/
There are a number of small Linux distributions out there but this one I think will be among the best maintained. Another thing is, if you need to grow it to full size, it's a trivial operation.
You should try Arch Linux. It let's you choose what you do and do not want to install on your system. You could even choose to not install bash :p Alternatively you could try Gentoo, where you have to compile everything from scratch.
Minimal Development Distro:
busybox, musl C library, gcc, binutils, make, bash, and the Linux kernel.
Minimal Non-Development Distro:
busybox, musl and the Linux Kernel
Minimal Linux Live is a small script that:
I have modified it to allow running it from the kernel source directory: https://github.com/cirosantilli/runlinuxx
It does not have
The ISO then leaves you in a minimal shell with busybox.
With QEMU you can easily boot into the system.
and you will be left inside a QEMU Window with you new minimal system. Awesome.