The things I know of and can think of off the top of my head are:
1. appimage https://appimage.org/
2. nix-bundle https://github.com/nix-community/nix-bundle
3. guix via guix pack
4. A small collection of random small projects hardly anyone uses for docker to do this (i.e. https://github.com/NilsIrl/dockerc )
5. A docker image (a package that runs everywhere, assuming a docker runtime is available)
7. https://en.wikipedia.org/wiki/Snap_(software)
AppImage is the closest to what you want I think.
But you can't take .so files and make one "static" binary out of them.
You can change the rpath though, which is sort of like an LD_LIBRARY_PATH baked into the object, which makes it relatively easy to bundle everything but libc with your binary.
edit: Mild correction, there is this: https://sourceforge.net/projects/statifier/ But the way this works is that it has the dynamic linker load everything (without ASLR / in a compact layout, presumably) and then dumps an image of the process. Everything else is just increasingly fancy ways of copying shared objects around and making ld.so prefer the bundled libraries.
https://appimage.github.io/appimagetool/
Myself, I've committed to using Lua for all my cross-platform development needs, and in that regard I find luastatic very, very useful ..
if you configure binfmt_misc
>Windows
if you disable Windows Defender
>OpenBSD
only older versions
Gave up them afterwards. If I need to tweak dependencies might as well deal with the packet manager of my distro.