Better: busybox build. It already has tar, xz and wget.
You just need to compile a simple HTTP client, or better:
find a gopher mirror. Bare netcat would work against that.
gopher://ftp.icm.edu.pl:70/1/vol/rzm6/busybox/binaries
With NetCat:
echo vol/rzm6/busybox/binaries/1.21.1/ | nc ftp.icm.edu.pl 70
echp vol/rzm6/busybox/binaries/1.21.1/busybox-i586 | nc ftp.icm.edu.pl 70 > busybox
Then ln all the binaries. Or run busybox wget.
Accesign the rest of the files is the same, just redirect the nc command to a file.
If tar is missing, you can just get for sure an static build and compile it. Or try to find an sltar mirror to github ;).
But that busybox has ustar, so look how easy it was with just netcat and Gopher, so writting an nc clone (openbsd's netcat portable version is really easy, altough there are another ones many more times dumbed down) would be the easiest win.
That is, that you can find them in an uncompressed form. If you fees adventurous you can probably coax the compressed data somehow through the bootloader decompressor, or the xz or deflate implementation found in the Linux kernel. Unfortunately these days modules tend to be compressed as well, but it should be possible to convice libdl to open a .ko file. If not you can always mmap it PROT_READ|PROT_EXEC and follow the symbols entries through the TLD yourself.
No nc.
https://github.com/kisom/goat/blob/master/goat.go
Well, JK. A simple, braindead, ugly and fast netcat clone (client only) can be done in C in few functions.
The exercise states that you have a Linux kernel and there is a network connection. If the kernel does not have TCP, but only offers a raw socket, well, then you have to write down a very simple TCP stack.
But if the kernel does have TCP, the exercise boils down to writing a very simple HTTP client, a very simple DNS resolver and a bare bones XZ decompressor and tar reader. From there I'd grab myself https://alpha.de.repo.voidlinux.org/static/xbps-static-lates... then execute
./xbps-install -S --repository=https://alpha.de.repo.voidlinux.org/current -r / base-system
and call it a day.