Introducing glibc will break your core binaries. Updated core binaries require a newer kernel which breaks hardware drivers (like NVIDIA) that need your old kernel. This "vicious cycle" makes it impossible to use new glibc w/o breaking your system.

The only solution is to compile the unsupported drivers somehow for the new kernel. So far this is not possible.

If you try using GLIBC without updating its dependencies, you will get complaints of a version mismatch on its dependencies which go all the way down to the kernel, which is why it is not possible.

Your options are limited and it is not possible unless you use a virtual machine. But likely VM is not what you want because VMs do not have the advantage of talking to your hardware directly. They have some VM extensions for making that better but they only cover CPU and RAM, not the video/sound card, that is all emulated.

Summary

If you want to run something that needs newer hardware, there is just no way around that:

1. You cant use unsupported hardware on newer kernel, and therefore linker, and then GLIBC.

2. You can't use unsupported GLIBC on older kernel which relies on the new kernel features. If you try to do it by force (point to new compiled version of the new binaries) you will get an error that the linker/kernel versions are incorrect.

3. The only solution to this is if there was a way to update your hardware drivers. If that isn't your problem, then UPGRADE UBUNTU to the latest version. If that IS your problem, then you are out of luck.

4. Companies like NVIDIA and AMD drop video support after a few years and leave you up a creek with no paddle, stranded on an old OS unless you buy yet another video card (even if your current one is fast and does just fine, they do NOT care).

5. Sandboxing in flatpak still needs those libraries to be able to link with your kernel. Sandboxing only fixes dependencies that are above the base system level. HOWEVER, if any of those libraries were built against a binaries that recursively rely on newer core libraries, you will still be stuck and it will still not work, below is an example:

/lib/x86_64-linux-gnu/libc.so.6: version `GLIBC_2.34' not found
(required by /tmp/.mount_my2newapp.a9fz3/usr/bin/../../usr/lib/liblzma.so.5)

As you can see here, even though the appimage/flatpak/snap was sandboxed, it still needed these new libraries further down the dependency chain.

I wouldn't install it, but unpack it in some work directory and refer to it via LD_LIBRARY_PATH. Downgrading your libc6 package can be quite problematic.

Your customer should be able to supply you with their libc6 version. It might, however, be advisable to use the libc6-dbg package which includes debug symbols. Unfortunately the 7u4 version is an older one and not available anymore.

Answer from @Dolphin isn't complete as it produces error from make: Makeconfig:42: *** missing separator. Stop.

In my case, to I had to do following:

# Check GLIBC_2.29
ldd --version | head -n1

# Build GLIBC_2.29 from sources
sudo apt-get install gawk bison -y
wget -c https://ftp.gnu.org/gnu/glibc/glibc-2.34.tar.gz
tar -zxvf glibc-2.34.tar.gz && cd glibc-2.34
mkdir glibc-build && cd glibc-build
../configure --prefix=/opt/glibc-2.34
make 
sudo make install

P.S. If you are trying to run ord just try building from sources, it's much simpler than upgrading ubuntu or recompiling GLIBC

You could build and install desired version of GLIBC from source into alternate location, and make the single application use it, as described in this answer.

Or you could prepare a suitable chroot jail, and run the application from it.

Or you could run the application inside of a virtual machine, such as Oracle VirtualBox.

For most security updates such as this you should be able to rest easy knowing that if your version of Ubuntu is still actively supported you will automatically receive such important updates.

Check in 'Software & Updates' that you have the correct boxes checked to:

1. Enable you to receive Security Updates
2. Allow regular checking of the Repository
3. Optionally automatically download and install Security Updates

Below is a screenshot showing you the relevant section of 'Software & Updates':

This screenshot is for Ubuntu 15.1 Wily Werewolf but will be the same through most modern releases of Ubuntu...

In my situation, the error appears when I try to run an application (compiled on Ubuntu 12.04 LTS) using GLIBC_2.14 on Debian Wheezy (which installs glibc 2.13 by default).

I use a tricky way to run it, and get correct result:

1. Download libc6 and libc6-dev from Ubuntu 12.04 LTS

2. Run dpkg command to install them into a directory (/home/user/fakeroot/ for example):

   $ dpkg -x libc6-dev_2.15-0ubuntu10.6_amd64.deb /home/user/fakeroot/
   $ dpkg -x libc6_2.15-0ubuntu10.6_amd64.deb /home/user/fakeroot/
   

3. Run your command with specified LD_LIBRARY_PATH:

   $ LD_LIBRARY_PATH=/home/user/fakeroot/lib/x86_64-linux-gnu/ YOUR_COMMAND
   

4. My application only uses memcpy() from GLIBC_2.14, and it works.

   I don't know whether it will work successfully for other applications.